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\title{\bf Arc-transitive dihedral regular covers of cubic graphs}

% input author, affilliation, address and support information as follows;
% the address should include the country, and does not have to include
% the street address 

\author{Jicheng Ma\thanks{Supported by CUAS grant R2012SC22 and Program for Innovation Team Building at Institutions of Higher Education in Chongqing (grant no. KJTD201321)}\\
\small Department of Mathematics \& KLDAIP\\[-0.8ex]
\small Chongqing University of Arts and Sciences\\[-0.8ex] 
\small Chongqing 402160, China.\\
\small\tt ma{\_}jicheng@hotmail.com}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Jan 16, 2014}{June 28, 2014}\\
\small Mathematics Subject Classifications: 05C25, 20B25}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
% should be informative, clear, and as complete as possible. Phrases
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\begin{abstract}
A regular covering projection is called $dihedral$ or $abelian$ if the covering transformation group is dihedral or abelian.
A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. 
In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs.
In particular, we classify all arc-transitive dihedral regular covers of $K_4$, $K_{3,3}$, the 3-cube $Q_3$ and the Petersen graph. 

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} Arc-transitive graph; Regular cover; Dihedral cover; Cubic graph
\end{abstract}

\section{Introduction}

Covering techniques are known to be a useful tool in algebraic and topological graph theory. Application of these techniques has resulted in many important examples and classification of certain families of graphs with particular symmetry properties. 
For example, Djokovi\' c used graph covers to prove that there exist infinitely many 5-arc-transitive cubic graphs, as elementary abelian covers of Tutte's 8-cage.  

Recently, quite a lot of attention has been paid  to the classification of arc-transitive covers of symmetric graphs. 
Approaches have involved voltage graph techniques (see \cite{Malnic}) and universal group methods (see \cite{ConderMa1}). 
In most cases, the group of covering transformation is either cyclic or elementary abelian, or more generally abelian. 
%
These methods have been successfully applied in the classification of arc-transitive elementary abelian or abelian covers of symmetric cubic graphs, such as the complete graph $K_4$, the complete bipartite graph $K_{3,3}$, the 3-cube graph $Q_3$, the Petersen graph and the Heawood graph and so on.  

In this paper, we are aiming to extend our research on arc-transitive abelian covers to non-abelian covers which is harder and has not been previously considered. 
We begin with some further background in Section 2,  and determine the arc-transitive cyclic regular covers of the M\" obius-Kantor graph and the Desargues graph in Sections 3 and 4, respectively. 
In Section 5, we deal with dihedral covers, and give a complete classification of arc-transitive dihedral covers of $K_4$, $K_{3,3}$, $Q_3$ and the Petersen graph. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\section{Preliminaries}\label{s-prelims}
 %covering graph, graph homomorphism locally bijective
%quotient graph, orbits of subgroup as vertices, and two vertices are connected if and only if the two $N$-orbits are connected. 
 %Cayley graph, normal cayley graph
 
 
 Throughout this paper, all the graphs are finite and simple. A $covering\ projection$ is defined as a graph homomorphism $p:  \tilde{X} \rightarrow X$ which is surjective and locally bijective, which means that the restriction $p: N(\tilde{v})\rightarrow N(v)$ is a bijection whenever $\tilde{v}$ is a vertex of $\tilde{X}$ such that $p(\tilde{v})= v\in V(X)$. 
We call $X$ the base graph, $\tilde{X}$ a covering graph. 
A covering  projection  $p:  \tilde{X} \rightarrow X$ is called $regular$ if there exists a semi-regular subgroup $N$ of the automorphism group ${\rm Aut}(\tilde{X})$ of $\tilde{X}$ such that the quotient graph $\tilde{X}/N$ (with vertices taken as the orbits of $N$, and two vertices adjacent whenever there exists an edge between these two $N$-orbits) is isomorphic to $X$. 
In that case we call $\tilde{X}$ a {\em regular cover} of $X$.
% and $N$ the {\em covering} ($transformation$) $group$, or {\em voltage group}. 
The regular covering projection is called {\em dihedral} (or $cyclic$) if $N$ is a dihedral (or cyclic) group. Similarly, we say a regular covering projection is $abelian$ (or {\em elementary abelian}) when the group $N$ is $abelian$ (or {\em elementary abelian}). 

Let $p: \tilde{X} \to X$ be a covering projection, and suppose 
$\alpha$ and $\beta$ are automorphisms of $X$ and $\tilde{X}$ such that 
$\alpha\,\circ\, p= p\,\circ\, \beta$, 
that is, such that the following diagram commutes: 
 
\begin{center}
\begin{tabular}{rcl}
 & $\beta$ & \\[-1mm]
 $\tilde{X}$ & $\longrightarrow$ &   $\tilde{X}$ \\[-3mm]
 &   & \\
$p \ \downarrow$ \ &  & \ $\downarrow \ p$ \\[-3mm]
 & & \\
 $\tilde{X}$ & $\longrightarrow$ & $X$ \\
 & $\alpha$ & \\
\end{tabular}
\end{center}
%
Then we say that $\alpha$ {\em lifts\/} along $p$ to $\beta$, and $\beta$ {\em projects\/} to $\alpha$, 
and also we call $\beta$ a {\em lift\/} of $\alpha$, and $\alpha$ a {\em projection\/} of $\beta$. 
Note that $\alpha$ is uniquely determined by $\beta$, but $\beta$ is not generally 
determined by $\alpha$. 
The set of all lifts of a given $\alpha\in{\rm Aut}(X)$ is denoted by $ \tilde{\alpha}$. 
If every automorphism of a subgroup $G$ of ${\rm Aut}(X)$ lifts,  
then $\bigcup_{\alpha\in G} \tilde{\alpha}$ is a subgroup of ${\rm Aut}(\tilde{X})$, called the {\em lift\/} of $G$.  
 
In particular, the lift of the identity subgroup of ${\rm Aut}(X)$ (or equivalently, the subgroup 
of all automorphisms of $\tilde{X}$ that project to the identity automorphism of $X$) is called 
the {\em group of covering transformations\/}, or {\em voltage group\/}, and is 
sometimes denoted by ${\rm CT}(p)$. 
%Also the regular cover $Y$ and the regular covering projection $p$ are called {\em abelian}, or {\em cyclic}, or {\em elementary abelian}, if the  group $N$ is abelian, or cyclic, or elementary abelian,  respectively. 
The normalizer of ${\rm CT}(p)$ in ${\rm Aut}(\tilde{X})$ projects to the largest subgroup of 
${\rm Aut}(X)$ that lifts.  Hence in particular, if the latter subgroup is $A$, say, then the 
lift of $A$ has a normal subgroup ${\rm CT}(p)$ with quotient  isomorphic to $A$. 
 
Two regular covering projections $p \!: Y \to X$ and $ p' \!: Y' \to X$ are called 
{\em isomorphic\/} if there exist graph isomorphism $\tilde{\theta} \!: Y \to Y'$ and graph automorphism $\theta: X\to X$
such that $\theta p= p' \tilde{\theta}$. 
In particular, isomorphic covering projections $p$ and $p'$ are called {\em equivalent}, if $\theta$ is the trivial automorphism. 
Similarly, two regular covers $Y$ and $Y'$ are called {\em equivalent} if the two regular covering projections $p$ and $p'$ are  equivalent. 
Usually, regular covers are studied up to equivalence. 

   
%%%%%%%%%
\smallskip
For every symmetric cubic graph, we now know that the automorphism group is a quotient of one of seven finitely-presented groups, which can be listed as $G_1, G_2^1, G_2^2, G_3$, $G_4^1, G_4^2$ and $G_5$, and presented as follows (see \cite{DjokovicMiller, ConderNedela})


%
\vskip 6pt \noindent 
\quad\quad$G_1  = \langle\, h,a \,|\ h^3 = a^2 = 1 \,\rangle$ ;
\vskip 6pt \noindent 
\quad\quad$G_2^{\,1}  = \langle\, h,p,a \,|\ h^3 = p^2 = a^2 = 1, \ 
php = h^{-1}, \ a^{-1}pa = p \,\rangle$;
\vskip 6pt \noindent 
\quad\quad$G_2^{\,2}  = \langle\, h,p,a \,|\ h^3 = p^2 = 1,\ a^2 = p,  \
php = h^{-1}, \ a^{-1}pa = p \,\rangle$;
\vskip 6pt \noindent 
\quad\quad$G_3    = \langle\, h,p,q,a \,|\ h^3 = p^2 = q^2 = a^2 = 1, \ 
pq = qp, \ php = h , \ qhq = h^{-1}, \ a^{-1}pa = q \,\rangle$;
\vskip 6pt \noindent 
\quad\quad$G_4^{\,1}  = \langle\, h,p,q,r,a \,|\ h^3 = p^2 = q^2 = r^2 = a^2 = 1, \ 
pq = qp, \ pr = rp, \ (qr)^2 = p, $ \vskip 2pt \hskip 95pt 
$h^{-1}ph = q, \ h^{-1}qh = pq, \ rhr = h^{-1}, \ a^{-1}pa = p, \
a^{-1}qa = r \,\rangle$;
\vskip 6pt \noindent 
\quad\quad$G_4^{\,2}  = \langle\, h,p,q,r,a \,|\ h^3 = p^2 = q^2 = r^2 = 1, \ a^2 = p, \ 
pq = qp, \ pr = rp, \ (qr)^2 = p, $ \vskip 2pt \hskip 84pt 
$h^{-1}ph = q, \ h^{-1}qh = pq, \ rhr = h^{-1}, \ a^{-1}pa = p, \
a^{-1}qa = r \,\rangle$;
\vskip 6pt \noindent 
\quad\quad $G_5  = \langle\, h,p,q,r,s,a \,|\ h^3 = p^2 = q^2 = r^2 = s^2 
= a^2 = 1, \ pq = qp, \ pr = rp, \ ps = sp, \ $  \vskip 2pt \hskip 105pt 
$qr = rq, \ qs = sq, \ (rs)^2 = pq, \ 
h^{-1}ph = p, \ h^{-1}qh = r, \ $ \vskip
2pt \hskip 105pt 
$h^{-1}rh = pqr, \ 
shs = h^{-1}, \ a^{-1}pa = q, \ a^{-1}ra = s \,\rangle$.

\vskip 6pt 
If a finite group $G$ acts as an $s$-arc-regular group of automorphisms of a cubic graph $X$, then $G$ is a smooth quotient of  $G_s$ or $G_s^{\,i}$,
where $i=1$ or $2$ depending on whether or not the group contains an involution 
$a$ that reverses an arc (in the cases where $s$ is even).  
(Note, `smooth' here means that the orders of the generators are preserved.)
Let $\cal U$ be either $G_s$ or $G_s^{\,i}$, then $G$ is a smooth quotient ${\cal U}/ K$ of $\cal U$ by some torsion-free normal subgroup $K$.  
If $\tilde{X}$ is a regular cover of $X$ admitting a group action of the same type, then there exists a normal subgroup $L$ of $\cal U$ contained in $K$, with ${\cal U}/ L$ being the corresponding group automorphisms of $\tilde{X}$. 
Then the group ${\cal U}/ L$ is an extension of the covering group $K/L$ by the given group $G= {\cal U}/ K$.
In order to find all cyclic covers, we need to find all possibilities for $L$ such that $K/L$ is cyclic. 
The presentation of $K$ can be found using Reidemeister-Schreier Theory, or by use of the {\tt Rewrite} command in {\sc Magma} \cite{Magma}. 
In the cases we will consider, $K$ is a free abelian group of finite rank $d$, namely the Betti number of the base graph $X$, with some basis $\{ w_1, w_2, \cdots, w_d\}$.
Algebraic or computational techniques can be applied to find the actions by conjugation of the generators of $\cal U$ on the generators of $K$. And  these actions induce linear transformations of the free abelian group $K$. Equivalently, a $d$-dimensional matrix representation of the group $G= {\cal U} / K$ can be given. 
Therefore, in order to find all the cyclic covers, we need to find  
all the $G$-invariant subgroups $L$ of rank $d-1$, equivalently we need to find all the $(d-1)$-dimensional representation of $G$.  

More details of Conder and the author's universal group method can be seen in \cite{ConderMa1} and \cite{Ma2}. 
Here we introduce some computational techniques that involves using {\sc Magma}.  
To find all finite cyclic regular covers with cyclic covering groups of exponent $m$, we may consider the action of $G$ by conjugation on the generators of $K/K^{(m)}$, where $K^{(m)}$ is the characteristic subgroup of $K$ generated by the $m$th powers of all elements of $K$. Since $K/K^{(m)}$ is $G$-invariant, we use {\sc Magma} to construct a finite group $K/K^{(m)}\rtimes G$ which is the extension group of $G$ by $K/K^{(m)}$. Note that this can be done, since both $K/K^{(m)}$ and $G$ are finitely-presented subgroups of $\cal U$. 
With a finite group stored in {\sc Magma} we can use the commands $\tt NormalSubgroups$ and $\tt meet$ to find all the subgroups $L$ of $K/K^{(m)}$ which are normal in $K/K^{(m)}\rtimes G$.  
Note that the `type' of group $K/K^{(m)}\rtimes G$ may not work for using the above commands in {\sc Magma}, then what one needs to do is using the {\em double coset graph} construction method (more details can be seen in \cite[Section 2]{ConderMa1}) to transform it into a permutation group (namely, an arc-transitive group of automorphisms). 
This method works successfully for `small' integer $m$. 
Generally, if $m= p_1^{e_1}p_2^{e_2}\cdots p_t^{e_t}$ is the prime-power factorisation of $m$ with distinct primes $p_i$, then the factor $K/L$ is a direct product of its Sylow subgroups. It follows that we need only consider the $G$-invariant subgroups of prime-power index in $K/K^{(m)}$. 

%In particular, if $m$ is prime, then we can use the following method. As in the previous paragraph, we have a $d$-dimensional matrix representation of group $G$ over prime field of characteristic $m$. Thus the {\sc Magma} commands $\tt GModules$ and $\tt Submodules$ work well for finding all the irreducible $G$-submodules. 

Once all the possibilities for $L$ have been found, we can determine additional information, such as uniqueness up to isomorphism and arc-transitivity of the covering graphs. 

 


%%%%%
 \section{Arc-transitive cyclic regular covers of the M\" obius-Kantor graph}
 
 
In this section, we classify all the arc-transitive cyclic  covering graphs of the M\" obius-Kantor graph $GP(8,3)$.
% that can be obtained by lifting the 1-arc-regular subgroup $GL(2, 3)$. 
The automorphism group of $GP(8,3)$ is isomorphic to ${\rm GL(2,3)}\rtimes C_2$ and acts 2-arc-regularly on the arcs. There are two other 1-arc-regular subgroups ${\rm GL(2,3)}$ and ${\rm SL(2,3)}\rtimes C_2$. 

Take the group $G_2^{\, 1}$, with presentation $G =\langle h, a, p\ |\ h^3= p^2= a^2=  (ph)^2= [a,p]=1\rangle $. The group $G_2^{\, 1}$ has two normal subgroups of index 96, both with quotient ${\rm GL(2,3)}\rtimes C_2$, but these are interchanged by the outer automorphism that takes the three generators $h,a$ and $p$ to $h, ap$ and $p$ respectively, so without loss of generality we can take either one of them. 

We will take the one that is contained in the subgroup $G_1= \langle h, a\rangle $; this is a normal subgroup $N$ of index 48 in $G_1$ with $G_1/ N\cong {\rm GL(2, 3)}$. 

Using %Reidemeister-Schreier theory (see \cite{Johnson} or \cite{LyndonSchupp}), or 
 the {\tt Rewrite} command in {\sc Magma}, we find that the subgroup $N$ 
is free of rank 9, on generators 

\begin{center}
\begin{tabular}{rclcrclcrclcrlc}
${w_1}  $&=&$(h^{-1} a h a h a)^2$ &  & 
${ w_2}$&=&$ (h^{-1} a  h  a  h^{-1}  a)^2$\\[+2pt]

${w_3} $&=&$(h^{-1}  a   h^{-1}  a  h  a)^2$ &&
${w_4}  $&=&$(h  a  h a  h^{-1}  a)^2$ \\[+2pt]

${w_5} $&=&$(h  a  h^{-1}   a  h  a)^2$ &  & 
${w_6}$&=&$(h  a  h^{-1}  a  h^{-1} a)^2$\\[+2pt]


${w_7} $&=&$h^{-1}  a  h^{-1}   a  h^{-1}  a h  a  h^{-1} a h^{-1} a h^{-1}$ &&
 ${w_8}$&=&$a  h a  h a  h  a  h  a  h^{-1} a   h^{-1}  a  h^{-1}  a h^{-1}$\\[+2pt]

  ${w_9}$&=&$a  h^{-1}  a  h^{-1}  a  h^{-1}  a  h  a h^{-1} a h^{-1} a  h^{-1} a$  \\

 
 \end{tabular}
\end{center} 

Easy calculations show that the generators $h, a$ and $p$ act by conjugation as below:  \\
(Note that the actions of generator $ap$ is just the composition of $a$ and $p$.)
%
\begin{center}
\begin{tabular}{rclcrclcrclcrlc}
${w_1}^ a $&=&${w_3}^{-1}$ &  & 
${ w_1}^h$&=&${w_7}^{-1} $ &   & 
${w_1}^ p $&=&$ w_6$   \\ 

${w_2}^ a $&=&${w_5}^{-1} $ &   & 
${w_2}^ h$&=&${w_2}^{-1}w_4$ &  & 
${w_2}^ p $&=&$ {w_5}$   \\ 

${w_3}^ a $&=&${w_1}^{-1}$ &  &
 ${w_3}^ h$&=&${w_3}^{-1}w_5$ &  &
  ${w_3}^p $&=&$w_4$   \\

 ${w_4}^ a $&=&${w_6}^{-1}$ &  &
  ${w_4}^ h$&=&${w_2}^{-1}$ &   & 
  ${w_4}^ p $&=&${w_3} $   \\
 
${w_5}^ a $&=&${w_2}^{-1}$ &  & 
${w_5}^ h$&=&${w_3}^{-1}$ &   & 
${w_5}^p $&=&${w_2} $   \\ 

${w_6}^ a $&=&${w_4}^{-1}$ &  &
 ${w_6}^ h$&=&${w_1}^{-1}$ &  &
  ${w_6}^ p$&=&${w_1}$   \\ 

${w_7}^a $&=&${w_9}$ &  &
 ${ w_7}^h$&=&${w_6}$ &   &
  ${w_7}^p$&=&$ {w_7}^{-1}$    \\ 
  
${w_8}^a $&=&${w_8}^{-1}$ &  &
 ${ w_8}^ h$&=&${w_1}{w_9}$ &   & 
  ${ w_8}^ p$&=&${w_7}^{-1}w_8{w_9}$ 
     \\ 
 
 ${w_9}^a $&=&${w_7}$ &  &
 ${ w_9}^ h$&=&${w_1}^{-1}{w_7}{w_8}^{-1}{w_9}^{-1}$&   & 
   ${ w_9}^ p$&=&${w_9}^{-1}$    \\ 
 \end{tabular}
\end{center} 
 

Now take the quotient $G_1/N'$ where $N'$ is the derived subgroup of $N$, which is an extension of the 
free abelian group $N/N' \cong \Z^9$ by the group $G_1 /N \cong {\rm GL(2,3)} $, 
and replace the generators $h, a$ and all $w_i$ by their images in this group. 
Also let $K$ denote the subgroup $N/N'$, and let $G$ be $G_1/N'$. 
Then, in particular, $G$ is an extension of $\rm GL(2,3) $ by $\Z^9$.    

By the above observations, we see that the generators $h,a$  and $p$ induce 
linear transformations of the free abelian group $K \cong \Z^9$ as follows: \\[-2pt] 
%
$$
a \mapsto \begin{pmatrix}
\phantom{-}0 &\phantom{-}0 & -1 & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 &\phantom{-}0  & \phantom{-}0& -1& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

-1 &\phantom{-}0 &\phantom{-}0  & \phantom{-}0&  \phantom{-}0 & \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 &\phantom{-}0  & \phantom{-}0& \phantom{-}0  &-1& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 & -1 &\phantom{-}0  & \phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\


\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & -1& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}1  \ \, \\

\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& \phantom{-}0& -1 &\phantom{-}0  \ \, \\


\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& \phantom{-}1&   \phantom{-}0 &\phantom{-}0  \ \, \\
 
\end{pmatrix},$$
$$
h \mapsto \begin{pmatrix}
\phantom{-}0 &\phantom{-}0 & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0& -1&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &-1 &\phantom{-}0  & \phantom{-}1&\phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 & -1  & \phantom{-}0&  \phantom{-}1 & \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 & -1 &\phantom{-}0  & \phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 & -1& \phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

-1 & \phantom{-}0  &\phantom{-}0  &\phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}1& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}1 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& \phantom{-}0& \phantom{-}0 &\phantom{-}1 \ \, \\

-1 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& \phantom{-}1&  -1 &-1  \ \, \\
 
\end{pmatrix},$$
and
 $$
p \mapsto \begin{pmatrix}
\phantom{-}0 &\phantom{-}0 & \phantom{-}0 & \phantom{-}0& \phantom{-}0& \phantom{-}1& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 &\phantom{-}0  & \phantom{-}0& \phantom{-}1& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 &\phantom{-}0  & \phantom{-}1&  \phantom{-}0 & \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}0 &\phantom{-}1  & \phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 &\phantom{-}1 &\phantom{-}0  & \phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\


\phantom{-}1 & \phantom{-}0  &\phantom{-}0  &\phantom{-}0& \phantom{-}0  &\phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& -1&\phantom{-}0 &\phantom{-}0  \ \, \\

\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  & \phantom{-}0& -1&\phantom{-}1 &\phantom{-}1  \ \, \\


\phantom{-}0 & \phantom{-}0  &\phantom{-}0  & \phantom{-}0 & \phantom{-}0  &\phantom{-}0& \phantom{-}0&   \phantom{-}0 &-1 \ \, \\
 
\end{pmatrix}.$$
 

 \bigskip
These matrices generate a group isomorphic to ${\rm Aut}( GP(8,3))$, with the first two generating a subgroup isomorphic to $\rm GL(2, 3)$; and the first and the product of the first and the third generating a subgroup isomorphic to ${\rm SL(2, 3)}\rtimes C_2$. 
Note that the matrices of orders 3, 2 and 8 representing $h, a$ and $ha$ have traces $-3, -1$ and $1$, 
respectively.   

Next, the character table of the group $\rm GL(2, 3)$ is given in Table~\ref{GL}, with $\gamma$ being the zeroes of the polynomial $t^2+2t+3$.

\begin{table}[!h]
\centering
\renewcommand\arraystretch{1.3}
\begin{tabular}{lcrrrrrrrrrrrrr}
\hline \\[-6mm]
%\vspace{1cm}
Element order  & \qquad  &  1 & 2 & 2 &3 & 4 & 6& 8 &8 \\
Class size  & \qquad  &  1 & 1& 12&  8&  6&  8&   6&   6 \\[+2pt]  
\hline \\[-11pt]
\qquad $\chi_1$ &  & $1$  & $1$  & $1$  & $1$  & $1$& $1$& $1$& $1$ \\  
\qquad $\chi_2$ &  & $1$& $1$  & $-1$ & $1$& $1$& $1$ & $-1$  & $-1$  \\  
\qquad $\chi_3$ &  & $2$  & $2$  & $0$  & $-1$  & $2$ & $-1$ & $0$ & $0$ \\  
\qquad $\chi_4$ &  &  $2$& $-2$&  $0$ &$-1$ & $0$ & $1$  & $\gamma$& $-\gamma$\\  
\qquad $\chi_5$ &  & $2$& $-2$&  $0$ &$-1$ & $0$ & $1$  & $-\gamma$& $\gamma$  \\
\qquad $\chi_6$ &  &  $3$& $3$&  $1$ &$0$ & $-1$ & $0$  & $-1$& $-1$\\ 
\qquad $\chi_7$ &  &  $3$& $3$&  $-1$ &$0$ & $-1$ & $0$  & $1$& $1$\\ 
\qquad $\chi_6$ &  &  $4$& $-4$&  $0$ &$1$ & $0$ & $-1$  & $0$& $0$\\[+3pt]  
 \hline
\end{tabular}
\caption{\small{The character table of the group $\rm GL(2, 3)$}}\label{GL}
\end{table}
 

By inspecting traces, we see that the character of the $9$-dimensional representation 
of $\rm GL(2, 3)$ over $\mathbb{Q}$ associated with the above action of $G = \langle h,a \rangle$ on $K$ 
is the character $\chi_3+\chi_4+\chi_5+\chi_7$, which is reducible 
to the sum of $\chi_3$, $ \chi_4+\chi_5$ and $\chi_7$, which are characters of three irreducible 2-, 4- and 3-dimensional representations over the rational field $\mathbb{Q}$. Especially, the 4-dimensional representation is reducible to two 2-dimensional irreducible representations over fields containing zeroes of the polynomial $t^2+2t+3$.
Therefore, for any prime $k$ other than 2 and 3, there is no $G$-invariant subgroup of rank 8. 

% In next two subsections, for every positive integer $m$ we let $K^{(m)}$ denote the subgroup of $K$ generated by the $m$th powers of all its elements, and if $m$ is a prime-power, say $m = k^{\ell}$, then we will consider $G_1$-invariant subgroups of each layer $K_{j+1}/K_j$ of $K/K_{\ell}$, where $K_j = K^{(k^j)}$ for every non-negative integer $j$.
 
% \subsection{Characteristic other than 2 and 3}
 
%\subsection{Characteristic 2 and 3}

For prime integers 2 and 3, with the help of {\sc Magma} by using the commands {\tt GModule} and {\tt Submodules} for matrix groups over prime fields, there is a unique $G$-invariant subgroup $U$ of rank 8, which is generated by 
 $w_1w_{9}, w_2w_{9}^{-1}, w_3w_{9}^{-1}, w_4w_{9}, w_5w_{9}, w_6w_{9}^{-1}, w_7w_{9}^{-1}$ and $w_8$ when $k= 3$. 
In particular, for prime integer 3 and exponent $3^2$, using the {\tt NormalSubgroups} command in {\sc Magma} we can show that 
there is no normal subgroup of rank 8 and exponent 9. 
%there is one normal subgroup of rank 8, which is generated by  $w_1w_{9}, w_2w_{9}^{-1}, w_3w_{9}^{-1}, w_4w_{9}, w_5w_{9}, w_6w_{9}^{-1}, w_7w_{9}^{-1}$ and $w_8$. 
 
Next, by calculation, we can see that the subgroup $U$ is also $p$-invariant for the additional generator $p$.
%$w_1w_{9} -- w_6w_9^{-1},  w_2w_{9}^{-1} -- w_5w_9, w_3w_{9}^{-1} -- w_4w_9, w_4w_{9} -- w_3w_9^{-1}, w_5w_{9} -- w_2w_9^{-1}, w_6w_{9}^{-1} -- w_1w_9, w_7w_{9}^{-1} -- w_7^{-1}w_9$ and $w_8 -- (w_7w_9^{-1})^{-1}w_8$. 
Hence the full automorphism group ${\rm GL(2, 3)}\rtimes C_2$ can be lifted, and the covering graph is at least 2-arc-transitive. 
 
 
 
\medskip
Now we consider the lifting of ${\rm SL(2,3)}\rtimes C_2$, which is  an 1-arc-regular subgroup generated by the cosets $Nh$ and $Nap$ of the quotient $G_1/ N$.

 
With the help of {\sc Magma}, a reduced character table of group $SL(2,3)\rtimes C_2$ is given in Table~\ref{SLC} where $\delta$ is a primitive 3rd root and $\phi$ is a primitive 4th root. 
 
Note that the traces of matrices induced by  $h$, $ap$, $hap, (haph)^2$ and $(haph)^3$ of orders 3, 2, 12, 6 and 4, respectively, are equal to $-3$, $-3$, 1, 1 and 1. 

 
\begin{table}[!h]
\centering
\renewcommand\arraystretch{1.3}
\begin{tabular}{lcrrrrrrrrrrrrr}
\hline \\[-6mm]
%\vspace{1cm}
Element order  & \qquad  &  1 & 2  & 3 & 4 & 6& 12  \\
Class size  & \qquad  &  1 & 6 & 4 & 6 &  4&  4 \\[+2pt]  
\hline \\[-11pt]
\qquad $\chi_1$ &  & $1$  & $1$  & $1$  & $1$  & $1$& $1$  \\  
 
\qquad $\chi_4$ &  &  $1$& $-1$&  $\delta$ &$1$ & $-1-\delta$ & $-\delta$  \\  
\qquad $\chi_5$ &  & $1$ & $-1$&  $-1-\delta$ &$1$ & $\delta$ & $1+\delta$   \\
\qquad $\chi_7$ &  &  $2$& $0$&  $-1$ &$0$ & $1$ & $-\phi$   \\ 
\qquad $\chi_8$ &  &  $2$& $0$&  $-1$ &$0$ & $1$ & $\phi$   \\ 
\qquad $\chi_{14}$ &  &  $3$& $-1$&  $0$ &$-1$ & $0$ & $0$   \\[+3pt]  
 \hline
\end{tabular}
\caption{\small{The character table of the group ${\rm SL(2, 3)}\rtimes C_2$}}\label{SLC}
\end{table}

Hence we can see that the character of the 9-dimensinal representation of ${\rm SL(2, 3)}\rtimes C_2$ over $\mathbb{Q}$ associated with the above action of $\langle h, ap \rangle$ on $K$ is $\chi_4+\chi_5+\chi_7+\chi_8+\chi_{14}$, which is reducible to the sum of $\chi_4+\chi_5$, $\chi_7+\chi_8$ and $\chi_{14}$. 
In particular, if there exists a primitive 3rd root $\delta$, then $\chi_4+\chi_5$ is reducible to $\chi_4$ and $\chi_5$; 
if there exists a primitive 4th root $\phi$, then $\chi_7+\chi_8$ is reducible to $\chi_7$ and $\chi_8$.  
 
Therefore, we can see that for prime $k\notin  \{ 2, 3\}$, if a primitive 3rd root $\delta$ exists, $K$ is a direct sum of four $G$-invariant subgroups of ranks 1, 1, 3 and 4; and if a primitive 4th root $\phi$ exists, $K$ is a direct sum of four $G$-invariant subgroups of ranks 2, 2, 2 and 3. 
%
(Note that, here we are only interested in the existence of $G$-invariant subgroups of rank 8.)
In fact, with the help of {\sc Magma}, if $\delta$ exists then these four $G$-invariant subgroups are generated by 
$\{w_1w_2^{\delta^2} w_3^{\delta}w_4 w_5^{\delta^2} w_6^{\delta} w_7w_8^{\delta^2}w_9^{-\delta} \}$, 
$\{w_1w_2^{\delta} w_3^{\delta^2}w_4 w_5^{\delta} w_6^{\delta^2} w_7w_8^{\delta}w_9^{-\delta^2} \}$, 
 $\{ w_1w_6^{-1}w_7^{-1}, \  w_2w_4^{-1}w_5w_6^{-1}w_8^{-1},\   w_3w_4^{-1}w_9\}$, and 
$\{ w_1w_6w_8, w_3w_4w_8$, $w_2w_4^{-1} w_5^{-1}w_6w_9^{-1},  w_7w_9^{-1} \}$, respectively. 
Especially, by the conjugation action of generator $a$, we can see that $a$ maps $w_1w_2^{\delta^2} w_3^{\delta}w_4 w_5^{\delta^2} w_6^{\delta} w_7w_8^{\delta^2}w_9^{-\delta}$ to $(w_1w_2^{\delta} w_3^{\delta^2}w_4 w_5^{\delta} w_6^{\delta^2} w_7w_8^{\delta}w_9^{-\delta^2})^{\delta}$. Hence these two cyclic covers are isomorphic, and we now take them as one cover. 
%Hence we have two cyclic covering groups which corresponding to  two cyclic covers. 
However, no cyclic covers exist for $k\notin \{2, 3\}$ when lifting the subgroup $\rm GL(2, 3)$. Therefore, the cyclic covering graph is 1-arc-transitive but not 2-arc-transitive. 

By \cite[Proposition 2.3]{ConderNedela}, this covering graph can not be 3-arc-transitive. 
Suppose this graph is 4-arc-transitive, then it is a cover of the Heawood graph, by \cite[Proposition 3.2]{ConderNedela}. 
%
Thus the cyclic covering group must be of order $7^e$ for some $e$. The full automorphism group of the cyclic cover  is  of order $16\cdot 8\cdot 7^e$. Since the 4-arc-transitive symmetric cubic graphs have vertex-stabilizer $S_4$, hence the order of  cyclic covering graph is equal to $16\cdot 8\cdot 7^e/ 24$ which is not an integer, contradiction. Therefore the cyclic covering graph cannot be 4-arc-transitive. 
Finally, again by \cite[Proposition 3.4]{ConderNedela}, if the covering graph is 5-arc-transitive, then it is a cover of the Biggs-Conway graph which is of order 2352. Similar to the above argument, the covering graph cannot be 5-arc-transitive. Therefore, these cyclic covering graphs are 1-arc-transitive.
 
 
  
For either $k$ equal to $2$ or $3$, similar to the lifting of $\rm GL(2, 3)$ with the help of {\sc Magma}, there is only one $G$-invariant subgroup of rank 8 of exponent $3$. %k= 3 
Hence not only the subgroup ${\rm SL(2, 3)}\rtimes C_2$ can be lifted but also the full automorphism group ${\rm Aut}(GP(8, 3))$ can be lifted. 
In particular, by Conder's list \cite{Conder1} we know that there is only one symmetric cubic graph of order 48; in which case the covering graph is 2-arc-regular. 
 
  
 %%
\begin{theorem}\label{MobiusKantor}  Let $n = k^e$ be any power of a prime $k$, with $e>0$. Then the arc-transitive cyclic regular covers of the M\" obius-Kantor graph with cyclic covering group of exponent $n$ are as follows: 

 $(1)$  For $k\equiv 1\ \mod\ 3$, 
 %all characteristic $k$ such that a primitive $3$rd root exists, 
 only the subgroup $SL(2, 3)\rtimes C_2$ can be lifted, and there is one $1$-arc-regular cover.
 
 $(2)$  For  $k= 3$ and $e= 1$, there is a unique $2$-arc-regular cover. 
\end{theorem}
   
 
 
 
 %%%%%
 \section{Arc-transitive cyclic regular covers of the Desargues graph}
 
In this section, we classify all the arc-transitive cyclic regular covering graphs of the Desargues  graph $GP(10, 3)$. 
The automorphism group of $GP(10, 3)$ is isomorphic to $S_5\times C_2$ and acts 3-arc-regularly on the arcs. There are two other 2-arc-regular subgroups $S_5$ and $A_5\times C_2$. 

Take the group $G_3$, with presentation $G =\langle h, a, p, q\ |\ h^3= p^2= q^2 = a^2=  (qh)^2= [p,q ] = [h, p]=[a,p]=apaq=1\rangle $. This group $G_3$ has a unique normal subgroup $N$ of index 240, with quotient $S_5\times C_2$. 

Using the {\tt Rewrite} command in {\sc Magma}, we find that the subgroup $N$ 
is free of rank 11, on generators \\[-30pt]

\begin{center}
\begin{tabular}{rclcrclcrclcrlc}
${w_1}  $&=&$ p  q  a  h  a  h^{-1}  a  h^{-1}  a  h  a  h  a  h^{-1}$&  & 
${ w_2}$&=&$ p  q  a  h^{-1}  a  h^{-1}  a  h  a  h  a  h^{-1}  a  h^{-1}$\\[+2pt]

${w_3} $&=&$ q  h  a  h  a  h^{-1}  a  h^{-1}  a  h  a  h  a  p$ &&
${w_4}  $&=&$ q  h  a  h^{-1}  a  h^{-1}  a  h  a  h  a  h^{-1}  a  p$ \\[+2pt]

${w_5} $&=&$ h  p  q  a  h  a  h^{-1}  a  h^{-1}  a  h  a  h  a  h$ &  & 
${w_6}$&=&$ h  p  q  a  h^{-1}  a  h^{-1}  a  h  a  h  a  h^{-1}  a  h$\\[+2pt]

${w_7} $&=&$ p  a  h  a  h^{-1}  a  h  a  h  a  h^{-1}  a  h^{-1}  a  h^{-1}  a  q$ &&
 ${w_8}$&=&$ q  a  h  a  h^{-1}  a  h  a  h  a  h^{-1}  a  h^{-1}  a  h^{-1}  a p$\\[+2pt]

${w_9}$&=&$ h  q  a  h  a  h^{-1}  a  h  a  h  a  h^{-1}  a  h^{-1}  a  h^{-1}  a  p  h^{-1}$ &&
  
 ${w_{10}}$&=&$ a  h  a  h  a  h  a  h  a  h  a  h^{-1}  a  h^{-1}  a  h^{-1}  a  h^{-1}  a  h^{-1}$ \\[+2pt]
  
${w_{11}}$&=&$ (a  h  a  h^{-1})^5$ 
 \end{tabular}
\end{center} 

Easy calculations show that the generators $h, a$ and $p$ act by conjugation 
as below: 
(Note that the action of $q$ can be given by the composition $apa$.)
%
\begin{center}
\begin{tabular}{rclcrclcrclcrlc}
${w_1}^ a $&=&${w_3}$ &  & 
${ w_1}^h$&=&${w_4}$ &   & 
${w_1}^ p $&=&$ w_2$   \\ 
${w_2}^ a $&=&${w_2}^{-1} $ &   & 
${w_2}^ h$&=&${w_3}$ &  & 
${w_2}^ p $&=&$ {w_1}$   \\ 
${w_3}^ a $&=&${w_1}$ &  &
 ${w_3}^ h$&=&${w_6}$ &  &
  ${w_3}^p $&=&$w_4$   \\
 ${w_4}^ a $&=&${w_4}^{-1}$ &  &
  ${w_4}^ h$&=&${w_5}$ &   & 
  ${w_4}^ p $&=&${w_3} $   \\
${w_5}^ a $&=&${w_7}^{-1}$ &  & 
${w_5}^ h$&=&${w_1}$ &   & 
${w_5}^p $&=&${w_6} $   \\ 
${w_6}^ a $&=&${w_8}$ &  &
 ${w_6}^ h$&=&${w_2}$ &  &
  ${w_6}^ p$&=&${w_5}$   \\ 
${w_7}^a $&=&${w_5}^{-1}$ &  &
 ${ w_7}^h$&=&${w_5}^{-1}w_6{w_9}^{-1}$ &   &
  ${w_7}^p$&=&$ {w_7}^{-1}$    \\ 
${w_8}^a $&=&${w_6}$ &  &
 ${ w_8}^h$&=&${w_3}{w_4}^{-1}{w_7}^{-1}$ &   & 
  ${ w_8}^ p$&=&${w_8}^{-1}$  \\ 
 ${w_9}^a $&=&${w_9w_{10}w_{11}^{-1}}$ &  &
 ${ w_9}^ h$&=&${w_8} $ &   & 
   ${ w_9}^ p$&=&${w_9}^{-1}$    \\ 
    ${w_{10}}^a $&=&${w_{10}^{-1}}$ &  &
 ${ w_{10}}^ h$&=&${w_3}{w_9}^{-1}{w_{10}}^{-1}{w_{11}}$&   & 
   ${ w_{10}}^ p$&=&${w_8}^{-1}w_9w_{10}$    \\ 
    ${w_{11}}^a $&=&${w_{11}^{-1}}$ &  &
 ${ w_{11}}^ h$&=&${w_6}^{-1}{w_8}{w_{10}}^{-1}$&   & 
   ${ w_{11}}^ p$&=&$w_7{w_9}^{-1}w_{11}$    \\ 
 \end{tabular}
\end{center} 
% and 
% \begin{center}
%\begin{tabular}{rclcrclcrclcrlc}
%${w_1}^ q $&=&${w_4}^{-1}$ &  & ${ w_2}^q$&=&${w_3}^{-1}$ &   & ${w_3}^ q $&=&$ {w_2}^{-1}$ &&${w_4}^ q $&=&${w_1}^{-1} $ \\
%${w_5}^ q$&=&${w_5}^{-1}$ &  & ${w_6}^ q$&=&$ {w_6}^{-1}$ &&${w_7}^ q $&=&${w_8}^{-1}$ &  & ${w_8}^ q$&=&${w_7}^{-1}$\\
% ${w_9}^q $&=&$w_5{w_6}^{-1}w_9$&& ${w_{10}}^q$&=&$w_6{w_{9}^{-1}}w_{11}$ &  &${w_{11}}^q $&=&$w_5w_9w_{10}$  
% \end{tabular}
%\end{center} 


Now take the quotient $G_3/N'$, which is an extension of the 
free abelian group $N/N' \cong \Z^{11}  $ by the group $G_3 /N \cong S_5\times C_2 $, 
and replace the generators $h, a, p$ and all $w_i$ by their images in this group. 
Also let $K$ denote the subgroup $N/N'$, and let $G$ be $G_3/N'$. 
Then, in particular, $G$ is an extension of $S_5\times C_2$ by $\Z^{11}$. 

By the above observations, we see that the generators $h, a$  and $p$ induce 
linear transformations of the free abelian group $K \cong \Z^{11}$ as follows:  
%
  $$
h \mapsto \left ( \begin{array}{ccccccccccc}
  \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}1& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
  \phantom{-}0 &\phantom{-}0 & \phantom{-}1  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
  \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}1& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}1& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
  \phantom{-}1 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
  \phantom{-}0 &\phantom{-}1 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
   \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& -1& \phantom{-}1& \phantom{-}0&\phantom{-}0 & -1 &\phantom{-}0&\phantom{-}0   \ \, \\
    \phantom{-}0 &\phantom{-}0 & \phantom{-}1  &  -1 & \phantom{-}0& \phantom{-}0&-1&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
     \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}1 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
      \phantom{-}0 &\phantom{-}0 & \phantom{-}1 & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &-1&-1&\phantom{-}1   \ \, \\
       \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& -1& \phantom{-}0&\phantom{-}1 &\phantom{-}0&-1&\phantom{-}0   
  \end{array}\right),$$
 %
 $$
a \mapsto \left ( \begin{array}{ccccccccccc}
\phantom{-}0 &\phantom{-}0 & \phantom{-}1  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
\phantom{-}0 &-1 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
\phantom{-}1 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & -1& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& -1&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}1 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& -1& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}1& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}1&\phantom{-}1& -1   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&-1&\phantom{-}0   \ \, \\
 \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0& -1   
 \end{array}\right),$$
 %%
 and 
 $$
p \mapsto \left ( \begin{array}{ccccccccccc}
  \phantom{-}0 &\phantom{-}1 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
    \phantom{-}1 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
      \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}1& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
        \phantom{-}0 &\phantom{-}0 & \phantom{-}1  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
          \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}1& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
            \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}1& \phantom{-}0& \phantom{-}0&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
              \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& -1&\phantom{-}0 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
                \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0& -1 &\phantom{-}0&\phantom{-}0&\phantom{-}0   \ \, \\
                  \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0&\phantom{-}0 &-1&\phantom{-}0&\phantom{-}0   \ \, \\
  \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}0& -1 &\phantom{-}1&\phantom{-}1&\phantom{-}0   \ \, \\
    \phantom{-}0 &\phantom{-}0 & \phantom{-}0  & \phantom{-}0& \phantom{-}0& \phantom{-}0& \phantom{-}1&\phantom{-}0 &-1&\phantom{-}0&\phantom{-}1  
   \end{array}\right).$$
  
 
These matrices generate a group isomorphic to $S_5\times C_2$, with the first two generating a subgroup isomorphic to $A_5\times C_2$; and the first and the product of the other two generating a subgroup isomorphic to $S_5$. 
Note that the matrices of orders 3, 2, 2, 6 and 6 representing $h, a, ap, hap$ and $(ha)^2h^{-1}a$ have traces $-1, -3, -1, 1$ and $1$, 
respectively.   
  
By inspecting traces and the character tables (which can be easily given by the {\tt CharacterTable} command in {\sc Magma}) of groups $A_5\times C_2$ and $S_5$, we see that the $11$-dimensional representation 
of $S_5$ over $\mathbb{Q}$ associated with the above action of $  \langle h,ap \rangle$ on $K$ 
is a sum of $U$ and $V$, which are two irreducible 6-dimensional  and 5-dimensional representations over the rational field $\mathbb{Q}$. 
Also the $11$-dimensional representation 
of $A_5\times C_2$ over $\mathbb{Q}$ associated with the action of $ \langle h,a \rangle$ on $K$ 
is a sum of $\varphi_1$ and $\varphi_2$, which are characters of two  irreducible 6-dimensional and 5-dimensional representations over the rational field $\mathbb{Q}$. However, in particular, if there exist zeros of the polynomial $t^2-t-1$, $\varphi_1$ is reducible to a sum of $\varphi_{1,1}$ and $\varphi_{1,2}$ each of which is a character of an irreducible 3-dimensional representation.  
 
Therefore, for any prime $k$ other than 2, 3 and 5, there is no $\langle h, ap \rangle$- and $\langle h, a \rangle$-invariant subgroup of rank 10; equivalently, no cyclic regular cover exists. 

For prime $k= 3$ and 5, with the help of {\sc Magma}, there is also  no $\langle h,ap \rangle$- and $\langle h,a \rangle$-invariant subgroup of rank 10. Thus there are no cyclic regular covers. 

For prime $k= 2$, with the help of {\sc Magma}, there are only two  $\langle  h,a \rangle$-invariant subgroups of rank 10 of exponent 2 and 4, respectively. Thus, correspondingly, there are two cyclic covering graphs of order 40 and 80. 
Also there are only two $\langle  h, ap \rangle$-invariant subgroups of rank 10 of exponent 2 and 4. 
By Conder's list \cite{Conder1}, we know that there are unique symmetric cubic graphs of orders 40 and 80, respectively, each of which is 3-arc-regular. Hence the above two cyclic covering graphs are exactly these two graphs.  
%And the full automorphism group $S_5\times C_2$ of the Desargues graph can be lifted. 
%Generated by $\{   w_1w_9^{-1},   w_2w_9,   w_3w_9,   w_4w_9^{-1},   w_5w_9^{-1},  w_6w_9,   w_7w_9^{-1},   w_8w_9^{-1}, w_{10}\} $
 
 
%%Desargues
\begin{theorem}\label{Desargues} There are only two  arc-transitive cyclic regular covers of the Desargues graph, both are $3$-arc-transitive, with cyclic covering groups $C_2$ and $C_4$, respectively. 
\end{theorem}
%\begin{proof}  \end{proof}
   

 
  
\section{Dihedral regular covers of cubic graphs}
\label{s-main} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

First of all, arc-transitive cubic graphs of small order like the complete graph $K_4$, the complete bipartite graph $K_{3,3}$, the 3-cube $Q_3$ and the Petersen graph are well known. The arc-transitive properties of each of the above graphs are as follows. 

The complete graph $K_4$ is $2$-arc-regular with automorphism group $S_4$, and the only arc-transitive subgroup of automorphisms of $K_4$ is  the subgroup $A_4$, which acts regularly on the arcs. 
%
The complete bipartite graph $K_{3,3}$ is $3$-arc-regular. Its  automorphism group is the wreath product $S_3\wr C_2$, and this contains three arc-transitive subgroups which act 1-, 2- and 2-arc-regularly on the arcs of $K_{3,3}$, respectively. 
In particular, two of these three subgroups are minimal, one is the group  $A_3\wr C_2$ which acts 1-arc-regularly, while the other is $(A_3\times A_3)\rtimes C_4$ which acts 2-arc-regularly. 
 %
The 3-cube $Q_3$ is 2-arc-regular, and its automorphism group is the direct product $S_4\times C_2$. And the only arc-transitive proper subgroups of automorphisms are $S_4$ and $A_4\times C_2$, each of which acts 1-arc-regularly on the arcs of $Q_3$. 
%
And finally, the Petersen graph is a 3-arc-regular graph. Its automorphism group is the symmetric group $S_5$, and the only other arc-transitive subgroup of automorphisms is the subgroup $A_5$, which acts 2-arc-regularly. 



Before investigating the dihedral covers, we remind readers of the following useful result given by Gardiner and Praeger in \cite{Praeger}. 
  \begin{theorem}{\em \cite{Praeger}}\label{Praeger} Let $\Gamma$ be a connected $G$-symmetric graph of valency $p$ a prime. For each normal subgroup $N$ of $G$ one of the following holds:
 
 (a) $\Gamma$ is $N$-symmetric;
 
 (b) $N$ acts regularly on vertices, so $\Gamma$ is a Cayley graph for $N$;
 
 (c) $N$ has just two orbits on vertices and  $\Gamma$ is bipartite; or 
 
(d) $N$ has $r\ge p+1$ orbits on vertices, the natural quotient graph $\Gamma_N$ on $N$-orbits is $G/N$-symmetric of valency $p$, and $\Gamma$ is a topological cover of $\Gamma_N$.
 \end{theorem}
 

Now, suppose graph $\tilde{X}$ is an arc-transitive dihedral regular $D_n$-cover of cubic graph $X$ where dihedral group $D_n$ is of degree $n$ (here, we always assume $n>2$), then we have the following lemma:

\begin{lemma}\label{lemma}
 $\tilde{X}$ is a cyclic regular cover of a $2$-cover of $X$. 
\end{lemma}
\begin{proof} Since $\tilde{X}$ is an arc-transitive dihedral cover of $X$, then there exists an arc-transitive subgroup $D_n\rtimes A$ of ${\rm Aut}(\tilde{X})$ which is the lifting subgroup of an arc-transitive subgroup $A$ of ${\rm Aut}(X)$. 
Let $C_n$ be the cyclic subgroup of $D_n$, then $C_n$ is normal in $D_n\rtimes A$. Especially, $C_n$ is a semi-regular subgroup of ${\rm Aut}(\tilde{X})$. Thus by Theorem~\ref{Praeger}, the quotient graph $\tilde{X}/ C_n$ is an arc-transitive 2-cover of $X$.  
\end{proof}





For the complete graph $K_4$, we have the following classification of all arc-transitive dihedral regular $D_n$-covers.

\begin{theorem}\label{k4}  For $n\ne 2$, graph $X$ is an arc-transitive dihedral regular $D_n$-cover of $K_4$ if and only if it is an arc-transitive cyclic regular $C_n$-cover of the 3-cube $Q_3$.
\end{theorem}





%Now, suppose graph $\tilde{X}$ is an arc-transitive dihedral regular cover of $K_4$ by lifting $A_4$, then the automorphism group of $X$ has an arc-transitive subgroup $D_n\rtimes A_4$ which is 
% (which is isomorphic to either $C_n\rtimes (C_2\times A_4)$ or $C_n\rtimes S_4$) 
%the lifting group of $A_4$. 
   
   
   
Before proving the above theorem, in  \cite{ConderMa1}, Conder and Ma gave the following results:
   
 \begin{theorem}{\rm \cite{ConderMa1}}\label{Q3}  Let $n = k^e$ be any power of a prime $k$, with $e>0$. Then the arc-transitive cyclic regular covers of the $3$-cube $Q_3$ with cyclic covering groups of exponent $n$ are as follows: 

 $(1)$  if $k\equiv 1\ \mod\ 3$, only the subgroup $A_4\times C_2$ can be lifted and there is one $1$-arc-regular cover.
 
 
 $(2)$  If $k= 3$ and $e= 1$, there is one $2$-arc-regular cover. 

 $(3)$  If $k= 2$ and $e= 1$, there is one $2$-arc-regular cover. 
\end{theorem}
%  
\smallskip
{\bf Proof of Theorem~\ref{k4}:}
By Lemma~\ref{lemma}, we know that $X$ is an arc-transitive cyclic cover of the $Q_3$ which is the only arc-transitive 2-cover of the $K_4$. 
%
From the above Theorem~\ref{Q3}, we know that there are only three types of cyclic covers. The first type, namely when $k\equiv 1\ \mod\ 3$, is of automorphism group $C_n\rtimes (A_4\times C_2)$. 
Note that from \cite[Section 6]{ConderMa1}, we know that $C_n\rtimes (A_4\times C_2)$ is generated by (the images of) elements $v_t$ and $h, ap$ where $\langle v_t\rangle \cong C_n$ and $\langle h, ap\rangle\cong  A_4\times C_2$. And also by \cite[Page 235, Paragraph 8]{ConderMa1} we have ${v_t}^h= {v_t}^t$ and ${v_t}^{ap}= {v_t}^{-1}$, hence ${v_t}^{(hap)^3}= {v_t}^{-1}$ where $(hap)^3$ is of order 2 and $\langle (hap)^3 \rangle \cong C_2$  which is normal in $ A_4\times C_2$. 
Therefore we have $C_n\rtimes (A_4\times C_2)\cong D_n\rtimes A_4$ which suggests that $X$ is a dihedral regular cover of $K_4$. 

For $k=3$, similarly, by \cite[Page 235, Paragraph -2]{ConderMa1}  the order 3 covering group is generated by $uv$, and  the conjugation action of $h$ and $ap$ are
$(uv)^h= {uv}$ and $(uv)^{ap}= (uv)^{-1}$. Thus $(uv)^{(hap)^3}= (uv)^{-1}$. 
Hence $C_3\rtimes (A_4\times C_2)\cong D_3\rtimes A_4$. (Note that the cyclic covering graph is of order 24, the structure of the automorphism group can also be easily checked by {\sc Magma}.)   %$\rightline
$\hfill\square$
%
 %$\hfill\blacksquare$
%$\blacksquare$
   
  
\begin{remark}  We know that there is a unique symmetric cubic graph of order 16 which is the M\" obius-Kantor graph. In particular, from {\rm \cite{ConderMa1}}, we know that it is a cyclic $C_4$-cover of the complete graph $K_4$ and also a $2$-cover of the $3$-cube $Q_3$.
\end{remark}
   
      
\begin{corollary}  Let $X$ be an arc-transitive dihedral regular  $D_n$-cover of $K_4$, then $n$ is of the following possibilities:

$(1)$ $3$ or $6$; or

$(2)$ $2^i3^jk^e$ for $i,j\in \{0, 1\}$ and prime integer $k\equiv 1\ \mod\ 3$ and $e>0$.\\[+2pt]
%
Note that the the product of integers $2, 3$ and $k^e$ is just the order of each cyclic covering group which comes from the direct product of cyclic groups $C_2$, $C_3$ and $C_{k^e}$. 
\end{corollary}
   
   
   
   
   %
About the complete graph $K_{3,3}$, we have the following result.    
   
\begin{theorem} The complete bipartite graph $K_{3,3}$ has no arc-transitive dihedral regular cover. 
\end{theorem}
\begin{proof}
Suppose $K_{3,3}$ has an arc-transitive dihedral regular cover $\cal D$, then by Lemma~\ref{lemma}, $\cal D$ is a cyclic regular cover of a 2-cover of $K_{3,3}$.
However, we know that $K_{3,3}$ has no arc-transitive 2-cover, and in fact, there is no arc-transitive cubic graph of order 12, contradiction. Hence $K_{3,3}$ has no arc-transitive dihedral covering graph.
%
\end{proof}

 
   
%Base on the Theorem~\ref{MobiusKantor}, 
For the 3-cube graph $Q_3$, the classification of arc-transitive dihedral covers is as follows. 
   
 \begin{theorem}
Let $X$ be an arc-transitive dihedral regular $D_n$-cover of the  $Q_3$, then $n$ is equal to $3$.
\end{theorem}
\begin{proof}
We know that each dihedral regular cover of $Q_3$ is a cyclic regular cover of the M\" obius-Kantor graph. In Theorem~\ref{MobiusKantor}, there are two types of cyclic regular covers of the M\" obius-Kantor graph. 
Firstly, 
if there exists a primitive 3rd root $\delta$, then the cyclic covering groups of the cyclic covers are generated by 
   $u= \{w_1w_2^{\delta^2} w_3^{\delta}w_4 w_5^{\delta^2} w_6^{\delta} w_7w_8^{\delta^2}w_9^{-\delta} \}$ and 
$v= \{w_1w_2^{\delta} w_3^{\delta^2}w_4 w_5^{\delta} w_6^{\delta^2} w_7w_8^{\delta}w_9^{-\delta^2} \}$, respectively. Since these two covering graphs are isomorphic, here we only consider the covering group generated by $u$.
The images of $u$ under the conjugation actions of generators $h$ and $ap$ are 
$u^4$ and $u^{-1}$. 
Since $SL(2, 3)\rtimes C_2 =\langle h, ap \rangle $, there is a unique normal subgroup of order 2 which is generated by $(haph)^6$. 
And the image of $u$ by the conjugation action of $(haph)^6$ is equal to $u^{16^6}$. Since $k\equiv 1 \ \mod\ 3$ but $16^6 +1\equiv 2\ \mod\ 3$. Hence $u^{16^6}\ne u^{-1}$, which suggests there is no dihedral normal subgroup of $SL(2, 3)\rtimes C_2 $. 

Secondly, for $k= 3$ and the cyclic covering group of order 3, the covering graph is 2-arc-regular and of order 48. With the help of {\sc Magma}, we can easily verify that its a dihedral regular covering graph of the $Q_3$, with automorphism group isomorphic to $D_3\rtimes (S_4\times C_2)$. 
 \end{proof}
   
  

In \cite{Ma1}, the author classified all the arc-transitive cyclic covers of the dodecahedron graph, and gave the following result.

  %%%
\begin{theorem}{\rm \cite{Ma1}}\label{dodecahedron}
Let $n = k^{\ell}$ be any power of a prime $k$, with $\ell>0$. 
Then the arc-transitive cyclic regular covers of the dodecahedron graph 
with covering group of exponent $n$ are as follows$\,:$
\begin{itemize} 
%
\item[{\rm (a)}]  If $k= 2$, there are exactly two such covers, 
namely  \\[+2pt]  
$\bullet$ one $3$-arc-transitive cover with covering group $\Z_2$ where $\ell =1$,  \\[+2pt]  
$\bullet$ one $3$-arc-transitive cover with covering group $\Z_4$ where $\ell =2$.

\item[{\rm (b)}]  If $k= 3$, there is exactly one such cover, 
namely  \\[+2pt]  
$\bullet$ one $2$-arc-transitive cover with covering group $\Z_3$ where $\ell =1$. %
 
 \item[{\rm (c)}]  There is no arc-transitive cyclic cover for other prime integer $k\ne 2, 3$.

 \end{itemize} 
\end{theorem}
     
 
 \begin{corollary}\label{2cover} All the arc-transitive cyclic regular covering graphs of the Desargues graph are also arc-transitive cyclic regular covers of the dodecahedron graph. 
 \end{corollary} 
 %%

Now, we can give the following results for arc-transitive dihedral regular $D_n$-covers of the  Petersen graph.


       
\begin{theorem}  Let $X$ be an arc-transitive dihedral regular $D_n$-cover of the Petersen graph, then $n$ is equal to either $3$ or $6$.
\end{theorem}
\begin{proof} 
First of all, every dihedral regular cover of the Petersen graph is a cyclic regular cover of a 2-cover of the Petersen graph. And we know that there are two 2-covers of the Petersen graph which are the dodecahedron graph and the Desargues graph.  However by Corollary~\ref{2cover}, we only need to consider the cyclic covers of the dodecahedron graph. 

By Theorem~\ref{dodecahedron}, we know that there are only finitely many cyclic covers. 
For $n=2$, by \cite{ConderMa1}, we know that the Petersen graph has a $(C_2)^2$-cover. 
For $n= 4$, the covering graph is of order 80 with automorphism group isomorphic to $Q_8\rtimes S_5$ where $Q_8$ is the quaternion group of order 8. Hence its a `quaternion' $Q_8$-cover of the Petersen graph instead of a dihedral cover.

  
%
Similarly, for $n= 3$, we have a 2-arc-transitive $C_3$-covering graph of automorphism group $C_3\rtimes (A_5\times C_2)$. With the help of {\sc Magma}, we have $C_3\rtimes (A_5\times C_2)\cong D_3\rtimes A_5$ which suggests that its a dihedral regular $D_3$-cover of the Petersen graph. 
 
Therefore, the Petersen graph only has two dihedral covers with covering groups $D_3$ and $D_6$.  
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
The author is indebted to the referee for the valuable comments and suggestions. And the author acknowledges the use of the {\sc Magma} computational 
package \cite{Magma}, which helped show many of the results given in this paper. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \bibliographystyle{plain} 
% \bibliography{myBibFile} 
% If you use BibTeX to create a bibliography
% then copy and past the contents of your .bbl file into your .tex file

\begin{thebibliography}{99}


\bibitem{Magma} 
W.~Bosma, J.~Cannon and C.~Playoust. 
The {\sc Magma} Algebra System I: The User Language. 
{\em J.~Symbolic Comput.}, 24:235--265, 1997. 
 %
 \bibitem{Conder1} 
M. D. E.~Conder.
Trivalent (cubic) symmetric graphs on up to 10000 vertices. \\
{\em http://www.math.auckland.ac.nz/$\sim$conder/symmcubic10000list.txt}, 2012.
%
\bibitem{ConderMa1} 
M.D.E.~Conder and J.~Ma. 
Arc-transitive abelian covers of cubic graphs. 
{\em J.~Algebra}, 387:215--242, 2013.   
%
\bibitem{ConderNedela} 
M.D.E.~Conder and R.~Nedela. 
A refined classification of symmetric cubic graphs. 
{\em J.~Algebra}, 322:722--740, 2009.   
%
\bibitem{DjokovicMiller} 
D.{\v Z}.~Djokovi{\'c} and G.L.~Miller. 
Regular groups of automorphisms of cubic graphs. 
{\em J.~Combin.~Theory~Ser. B}, 29:195--230, 1980.   
%
\bibitem{Praeger} 
A. Gardiner and C. E. Praeger.
On 4-valent symmetric graphs. 
{\em Europ. J. Combin.}, 15:375--381, 1994.   
%
\bibitem{Ma1} 
J.~Ma. 
On 3-arc-transitive covers of the Dodecahedron graph.
{\em preprint}.   
\bibitem{Ma2} 
J.~Ma, 
Arc-transitive abelian regular covering graphs.
{\em preprint}.   
%
\bibitem{Malnic} 
A. Malni\v c, D. Maru\v si\v c and P. Poto\v cnik. Elementary abelian covers of graphs. {\em J. Algebraic Combin.}, 20:71--97, 2004.
\end{thebibliography}  
 \end{document}