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

\title{On arc-transitive metacyclic covers of graphs\\ with order twice a prime}

% Input author, affiliation, address and support information as follows;
% The address should include the country, but does not have to include
%    the street address. Give at least one email address.

\author{Zhaohong Huang\thanks{Supported by Training foundation of Shandong Province (ZR2018PA005).}\\
\small School of Mathematics and Statistics Science\\[-0.8ex]
\small Ludong University\\[-0.8ex]
\small Yantai,
P. R. China\\
\small\tt 1328897542@qq.com\\
\and
Jiangmin Pan\thanks{Corresponding author; supported by
National Natural Science Foundation of China (11461007, 11231008)}\\
\small School of Statistics and Mathematics\\[-0.8ex]
\small Yunnan University of Finance and Economics\\[-0.8ex]
\small Kunming,
P. R. China\\
\small\tt jmpan@ynu.edu.cn}

\begin{document}

\maketitle

\begin{abstract}
Quite a lot of attention has been paid recently
to the characterization and construction of
edge- or arc-transitive abelian (mostly cyclic or elementary abelian)
covers of symmetric graphs,
but there are rare results for nonabelian covers
since the voltage graph techniques are generally not easy to be used in this case.
In this paper, we will classify certain metacyclic arc-transitive
covers of all non-complete symmetric graphs with prime valency and
twice a prime order $2p$ (involving the complete bipartite graph $\K_{p,p}$,
the Petersen graph, the Heawood graph, the Hadamard design on $22$ points
and an infinite family of prime-valent arc-regular graphs of dihedral groups).
A few previous results are extended.
\end{abstract}

\section{Introduction}

Throughout the paper, by a graph, we mean a connected,
simple and undirected graph with valency at least three.

For a graph $\Ga$ and an automorphism group $X\le\Aut\Ga$,
$\Ga$ is called $X$-{\it vertex-transitive},
$X$-{\it edge-transitive}
or $X$-{\it arc-transitive},
if $X$ is transitive on its vertex set, edge set or arc set, respectively.
For a positive integer $s$, an $s$-{\it arc} of $\Ga$ is
a sequence $\a_0,\a_1,\dots,\a_s$ of $s+1$ vertices such that
$\a_{i-1},\a_i$ are adjacent for $1\le i\le s$ and $\a_{i-1}\not=\a_{i+1}$
for $1\le i\le s-1$. Then
$\Ga$ is called {\it $(X,s)$-arc-transitive}
or {\it $(X,s)$-arc-regular}
if $X$ is transitive or regular on the set of $s$-arcs of $\Ga$,
respectively.
In particular, if $\Aut\Ga$ is regular on the set of $s$-arcs of $\Ga$,
then $\Ga$ is simply called {\it $s$-arc-regular}.

An essential method for studying edge- or arc-transitive graphs is taking normal quotient
graphs.
Let $\Ga$ be a graph with vertex set $V\Ga$,
and suppose that $X\le\Aut\Ga$ acts edge- or arc-transitively on $\Ga$
and has an intransitive normal subgroup $N$.
Denote by $V\Ga_N$
the set of $N$-orbits on $V\Ga$. Then the {\it normal quotient graph}
of $\Ga$ induced by $N$, denoted by $\Ga_N$, is defined  with
vertex set $V\Ga_N$ and two vertices $B,C\in V\Ga_N$ are adjacent if and
only if some vertex in $B$ is adjacent in $\Ga$ to some vertex in
$C$. If further, for each adjacent $B$ and $C$,
the induced subgraph $[B,C]\cong n\K_2$ is a complete matching,
where $n=|B|=|C|$, then $\Ga$
is called a {\it normal cover} (or {\it regular cover}, or {\it cover} for short) of $\Ga_N$.
In some cases, to emphasize the groups $X$ and $N$,
$\Ga$ is called an $X$-edge-transitive or $X$-arc-transitive $N$-cover
of $\Ga_N$ respectively.
If $N$ is a cyclic, abelian or nonabelian group,
then $\Ga$ is called a {\it cyclic cover},
{\it abelian cover} or {\it nonabelian cover} of $\Ga_N$,
respectively.
We remark that `the cover of graph' can also be defined by using
notions of fibre-preserving group and covering transformation group,
refer to \cite{Djokovic,Malnic-1,Malnic-2}.

From the above definition, an important strategy for studying
transitive graphs naturally arises, involving
the following two steps.
Step 1 would be concerned to obtain a characterization of
`basic' transitive graphs (that is, the graphs which are
not covers of their normal quotient graphs).
Step 2 would then consists in approaching all transitive covers of basic graphs.

Characterizing covers of graphs is thus often a key step
for studying edge- or arc-transitive graphs.
By using voltage graph techniques (which generally are powerful for
finding cyclic and elementary abelian covers, refer to \cite{GT,Malnic-1,Malnic-2,Skoviera}),
a lot of classifications of transitive cyclic or elementary abelian covers
of symmetric graphs have been obtained, for example, see
\cite{Feng-1,Malnic-2,MMP,Malnic-3} and references therein for edge- or arc-transitive
cyclic and elementary abelian covers of
certain small graphs,
and see \cite{DMW,DKX,DX16,Xu-Du,XD16} for 2-arc-transitive cyclic and
certain elementary abelian covers of $\K_n$ (the compete graph) and
$\K_{n,n}-n\K_2$ (the bipartite complete graph with a $1$-factor removed),
see also \cite{XD15}  for 2-arc-transitive metacyclic covers of $\K_n$.
Quite recently, a new approach was founded by investigating the action of
universal group of basic graph
and then used to find all arc-transitive abelian covers
of a few  small cubic symmetric graphs
(see \cite{Conder-1,Conder-2}).
However, since the voltage graph techniques are generally difficult
for treating nonabelian covers,
the results of nonabelian covers are rare.

The main purpose of this paper is to determine all arc-transitive
$K$-covers of graph $\Sig$,
where $\Sig$ is a non-complete graph with prime valency $r$ and twice a prime order $2p$,
and $K\cong\ZZ_m:\ZZ_q$ is a split metacyclic but not cyclic group
(the arc-transitive cyclic covers of $\Sig$
have been determined by \cite{PHD}), with $q$ a prime less than $r$.
We notice that certain special cases have been investigated
in the literature:
see \cite{Feng} for $K$ cyclic and $\Sig=\K_{3,3}$ ($p=3$ and $r=3$);
\cite{Feng-Kwak} for $K$ cyclic and $\Sig=\O_2$ the Petersen graph
($p=5$ and $r=3$);
\cite{Wang} for $K$ cyclic and $\Sig$ the Heawood graph ($p=7$ and $r=3$);
\cite{Pan-Huang} for $K$ cyclic and $\Sig=\K_{p,p}$ the complete bipartite graph with $p>3$;
and see \cite{Zhou-Feng} for $K$ a dihedral group and $r=3$.

The terminology and notation used in this paper are standard.
For example, for a positive integer $n$,
we denote by $\ZZ_n$, $\D_{n}$ with $n$ even, $\A_n$ and $\S_n$
the cyclic group and the dihedral group of order $n$,
the alternating and the symmetric group of degree $n$, respectively.
For two groups $N$ and $H$, we denote by $N\times H$ the direct product of $N$ and $H$,
by $N.H$ an extension of $N$ by $H$, and
if such an extension is split, then we write $N:H$ instead of $N.H$.
Also, we denote by $\F_n$, $\F_n\A$, $\F_n\B$ etc. with $n$ a positive integer
the corresponding cubic graphs of order $n$ in the Foster census,
see \cite{Bouwer} or \cite{Cub-768}.

The main result of this paper is the following theorem.
For convenience, the graphs appearing in the theorem which are not explained above
are introduced in Section 2.


\begin{theorem}
  \label{Thm-1}
Let $\Ga$ be an $X$-arc-transitive $K$-cover of a graph $\Sig$,
where $X\le\Aut\Ga$, $\Sig$ is a non-complete graph
of odd prime valency $r$ and order $2p$ with $p$ a prime,
and $K\cong\ZZ_m:\ZZ_q$ is a metacyclic but not cyclic group with $q<r$ a prime.
Then one of the following statements holds.
 \begin{itemize}
\item[(1)] $\Sig\cong\K_{3,3}$,
$K\cong\ZZ_{2n}\times\ZZ_2$ with $n$ odd,
and $\Ga$ is characterized in \cite[Theorem 5.1]{Conder-1};
\item[(2)] $\Sig\cong\CD(2p,3)$ with $p\equiv 1~(\mod 3)$,
and either
\begin{itemize}
\item[(i)] $K\cong\ZZ_2^2$ and $\Ga\cong\CGD(8p,3)$ is $1$-arc-regular.
or
\item[(ii)]
$K\cong\ZZ_{m}\times\ZZ_2$, and $\Ga\cong\CGD(2,{mp\over 2},\lambda)$,
or $\CGD(2p,{m\over 2p},\lambda)$
with $p\mid m$, where
$m=2\cdot3^sp_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}\ge 6$, $s\le 1$,
$0\le t$ and $p_1,p_2,\dots,p_t$ are distinct primes such that $3\mid (p_i-1)$ for $i=1,2,\dots,t$.
\end{itemize}

\item[(3)] $\Sig\cong\O_2$, and the tuple $(\Ga,K)$ is listed in the following table.

\[\begin{array}{llll}\hline

\Row & \Ga &  K & s-{\arc}-{\regular} \\ \hline

1 & \F40 & \D_4 & 3-\arc-\regular  \\

2 & \F60 & \D_6 &  2-\arc-\regular \\

3 & \F120\B & \D_{12} &  2-\arc-\regular \\
\hline
\end{array}\]

\end{itemize}
\end{theorem}

\vskip0.2in
The structure of this paper as follows. We give some preliminary results and introduce examples
that appear in Theorems~\ref{Thm-1}
in Section 2, and prove some technical lemmas in Section 3.
Then, we classify the mentioned covers of the complete bipartite graph $\K_{p,p}$
and graphs $\CD(2p,r)$ in Sections 3 and 4 respectively,
and complete the proof of
Theorem~\ref{Thm-1} in Sections 5.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries and examples}

\subsection{Preliminary results}
For a group $G$ and its subgroup $H$,
let $C_G(H)$ and $N_G(H)$ denote the centralizer and normalizer of $H$ in $G$, respectively.

\begin{lemma}\label{N/C}{\rm(\cite[Ch. I, Lemma 4.5]{Huppert})}
Let $G$ be a group and $H$ a subgroup of $G$.
Then $N_G(H)/C_G(H)\le\Aut(H)$.
\end{lemma}

A graph $\Ga$ is called a {\it Cayley graph} of  a group $G$
if there is a subset $S\subseteq G\setminus\{1\}$,
with $S=S^{-1}:=\{g^{-1}\mid
g\in S\}$, such that $V\Ga=G$
and two vertices $g$ and $h$ are adjacent if and
only if $hg^{-1}\in S$. This Cayley graph is denoted by $\Cay(G,S)$.
It is well known that a graph $\Sig$ is isomorphic to a Cayley graph
of a group $R$ if and only if $\Aut\Sig$
contains a subgroup which is isomorphic to $R$ and acts regularly on
$V\Sig$, see \cite[Proposition\,16.3]{Biggs}.
If this regular subgroup is normal in $X$ with $X\le\Aut\Sig$, then
$\Sig$ is called an $X$-{\it normal
Cayley graph}.
In particular, if this regular subgroup is normal in $\Aut\Sig$, then
$\Sig$ is called a {\it normal Cayley graph}.

Let $\Ga=\Cay(G,S)$. Let
$$\hat G=\{\hat g\mid \hat g:~x\mapsto xg,\ \mbox{for all $g,x\in G$}\},$$
$$\Aut(G,S)=\{\s\in\Aut(G)\mid S^\s=S\}.$$
Then both $\hat G$ and $\Aut(G,S)$ are subgroups of $\Aut\Ga$.
Further, the following nice property holds.

\begin{lemma}\label{N(G)}{\rm(\cite[Lemma 2.1]{Godsil})}
Let $\Ga=\Cay(G,S)$ be a Cayley graph. Then the normalizer
$N_{\Aut\Ga}(\hat{G})=\hat{G}:\Aut(G,S)$.
\end{lemma}

We remark that $\hat G$ is the right regular representation of $G$,
and so isomorphic to $G$.
For convenience, we will often
write $\hat G$ as $G$.

Let $H$ be a group acting transitively on a set $\Ome$.
Then $H$ is called {\it primitive} on $\Ome$
if $H$ preserves no nontrivial partition of $\Ome$.
A graph $\Ga$ is called {\it $X$-locally-primitive} with $X\le\Aut\Ga$,
if the vertex stabilizer $X_{\a}{:}=\{ x\in X\mid \a^x=\a\}$
acts primitively on the neighbor set $\Ga(\a)$ for each vertex $\a$.
Clearly, an edge-transitive graph with odd prime valency is locally-primitive.
By Lemma~\ref{N(G)},
one easily has the following assertion.

\begin{lemma}\label{Nor-CGs}
Let $\Ga=\Cay(G,S)$ be an $X$-normal locally-primitive Cayley graph of a
group $G$, where $G\lhd X\le\Aut\Ga$. Then
$X_{\bf 1}\le\Aut(G,S)$ and elements in $S$ are involutions,
where ${\bf 1}$ denotes the
vertex of $\Ga$ corresponding to the identity element of $G$.
In particular, if $G$ is abelian, then
$G$ is an elementary abelian $2$-group.
\end{lemma}

The following theorem provides a basic method for studying
vertex-transitive locally-primitive graphs
(see \cite{GLP} for the vertex-intransitive case), where parts (1--3)
were first proved by Praeger \cite[Theorem 4.1]{Praeger92}
for 2-arc-transitive graphs
and slightly generalized to locally-primitive graphs by \cite[Lemma 2.5]{Li-Pan};
part (4) follows easily by part (1).

\begin{theorem}\label{Praeger} Let $\Ga$ be an $X$-vertex-transitive
locally-primitive graph, and let $N\lhd X$ have at least three orbits on
$V\Ga$. Then the following statements hold.
\begin{itemize}
\item[(1)] $N$ is semiregular on $V\Ga$, $X/N\le\Aut\Ga_N$,
$\Ga_N$ is $X/N$-locally-primitive, and $\Ga$ is an $X$-locally-primitive $N$-cover
of $\Ga_N$.
\item[(2)] $\Ga$ is $(X,s)$-arc-transitive
if and only if $\Ga_N$ is $(X/N,s)$-arc-transitive,
where $1\le s\le 5$ or $s=7$.
\item[(3)] $X_{\a}\cong(X/N)_{\d}$, where $\a\in V\Ga$ and $\d\in V\Ga_N$.
\item[(4)] If $X$ has a normal subgroup $M$ which is contained in $N$,
then $\Ga_M$ is an $X/M$-locally-primitive $N/M$-cover of $\Ga_N$.
\end{itemize}
\end{theorem}

Given a group $G$ and two subgroups $L,R$ such that $L\cap R$
is {\it core-free} in $G$ (that is, $L\cap R$ contains no nontrivial normal subgroup of $G$).
Define a bipartite graph $\Ga=\Cos(G,L,R)$, called {\it bi-coset graph},
with vertex set $[G:L]\cup[G:R]$ and
$$\{Lx,Ry\}~is~an~edge~in~\Ga\iff yx^{-1}\in RL.$$
The following lemma gives some basic properties of the bi-coset graphs,
refer to \cite[Lemmas 3.5, 3.7]{GLP}.

\begin{lemma}\label{GLP}
With notation above,
$\Ga$ satisfies the following properties:
\begin{itemize}
\item[(1)] $\Ga$ is connected if and only if $\l L,R\r=G$;
\item[(2)] $G\le\Aut\Ga$, and $\Ga$ is $G$-edge-transitive and $G$-vertex-intransitive;
\item[(3)] $\Ga(L)=\{Rx\mid x\in L\}$ and $\Ga(R)=\{Lx\mid x\in R\}$;
\item[(4)] $|\Ga(\a)|=|L:L\cap R|$ and $|\Ga(\b)|=|R:L\cap R|$, where $\a\in [G:L]$
and $\b\in [G:R]$.
\end{itemize}

Conversely, each $G$-vertex-intransitive and $G$-edge-transitive graph
is isomorphic to\\
$\Cos(G,G_{\a},G_{\b})$, where $\a$ and $\b$ are adjacent vertices.
\end{lemma}


A group $G$ is called perfect if $G=G'$, where $G'$ is the commutator subgroup of $G$; and an
extension $G=N.H$ is called a {\it central extension}
if $N$ is contained in the centre $Z(G)$ of $G$.
If a group $G$ is perfect and $G/Z(G)$ is isomorphic to a simple group $T$,
then $G$ is called a {\it covering group} of $T$.
Schur \cite{Schur} showed that a simple (and more generally,
perfect) group $T$ possesses a ¡®universal¡¯ covering group $G$ with the property that every
covering group of $T$ is a homomorphic image of $G$; in this case, the center $Z(G)$ is
called the Schur multiplier of $T$, refer to \cite[P.43]{Gorenstein}.
The Schur multipliers of all simple
groups are known (\cite[P. 302--303]{Gorenstein}).
The following lemma is known.

\begin{lemma}\label{cent-ext}
%{\rm(\cite[Lemma 2.11]{Pan-2})}
Let $G=N.T$, where $N$ is cyclic or isomorphic to $\ZZ_p^2$ with $p$ a prime,
and $T$ is a nonabelian simple group. Then $G=N.T$ is a central extension,
$G=NG'$, and $G'=M.T$ is a perfect group, where
$M$ is a subgroup of both $N$ and the Schur Multiplier of $T$.
\end{lemma}

The next lemma gives an observation regarding
the vertex stabilizer of permutation groups,
that will be used later.

\begin{lemma}\label{Stabilizer}{\rm(\cite[Lemma 3.4]{PHD})}
Suppose that $G=NH$ is a permutation group on a set $\Ome$, where $N\lhd G$
and $H\le G$.
Then $G_{\a}\cong N_{\a}.o$ for $\a\in\Ome$, where $o\le H/(H\cap N)$.
In particular, if $N$ is transitive on $\Ome$, then $o\cong H/(H\cap N)$.
\end{lemma}

\subsection{Examples}
We now introduce examples appearing in Theorem~\ref{Thm-1}.

The first family of examples arises from Cayley graphs of dihedral groups,
stated in Example~\ref{normal-dih},
where the first two letters $\CD$ of the notation $\CD(2p,r)$
stand for `Cayley graph of a dihedral group'.

\begin{example}\label{normal-dih}
Let $G=\l a,b\mid a^p=b^2=1,a^b=a^{-1}\r\cong\D_{2p}$ be a dihedral group,
with $p$ an odd prime. Let $r$ be an odd prime
and $k$ a solution of the congruence equation
$$ x^{r-1}+x^{r-2}+\cdots + x+1\equiv 0~(\mod~p).$$
Define
$$\CD(2p,r)=\Cay(G,\{ b,ab,a^{k+1}b,\dots,a^{k^{r-2}+k^{r-3}+\cdots+1}b\}).$$

\end{example}
\vskip0.1in

We give some remarks here.
\begin{itemize}
\item[(a)] The congruence equation above has a solution if and only if
$p=r$ or $r\mid (p-1)$,
see \cite[Lemma 3.3]{Feng-Li}.

\item[(b)] The graph $\CD(2p,r)$ is a bipartite graph,
and up to isomorphism, it
is independent  of the choice of $k$ by \cite[Corollary 3.2]{Feng-Li}.
This is the reason why the notation used does not involve $k$.

\item[(c)] The graph $\CD(2p,p)\cong\K_{p,p}$,
$\CD(14,3)$ is the Heawood graph,
and $\CD(22,5)$ is the incidence graph of valency 5 (the another one is of valency 6)
of the Hadamard design on $22$ points.
\end{itemize}

A group $G$ is called a {\it generalized dihedral group},
if $G=H:\l a\r$ for some abelian subgroup $H$ and an involution $a$
such that $h^a=h^{-1}$ for each $h\in H$.
This group is denoted by $\Dih(H)$. Obviously, $\Dih(\ZZ_n)\cong\D_{2n}$.
The next family of examples arises from Cayley graphs of generalized dihedral groups,
which was first obtained in \cite{Zhou-Feng}.

\begin{example}\label{Cay-GenDih}
Let $m$ and $k$ be positive integers.
Let
$$\Dih(\ZZ_{mk}\times\ZZ_m)=\l a,b,c\mid a^2=b^{mk}=c^m=1,b^a=b^{-1},c^a=c^{-1},bc=cb\r$$
be a generalized dihedral group. Assume that $\lambda=0$ for $k=1$,
and $\lambda^2+\lambda+1\equiv 0~(\mod k)$ for $k>1$.
Define $$\CGD(m,k,\lambda)=\Cay(\Dih(\ZZ_{mk}\times\ZZ_m),\{a,ab,ab^{\lambda}c\}).$$
\end{example}

The first three letters $\CGD$ of the graph $\CGD(m,k,\lambda)$ stand for `Cayley
graph of a generalized dihedral group'.
By the notice before \cite[Theorem 3.1]{Zhou-Feng}, up to isomorphism,
the cubic graph $\CGD(2,p,\lambda)$
with order $8p$ is independent of the choice of $\lambda$,
 we thus denote it simply by $\CGD(8p,3)$.


The next lemma is quoted from \cite{C-O}, which classifies arc-transitive prime-valent graphs of
order twice a prime.

\begin{lemma}\label{2p-graphs}
Suppose that $\Sig$ is a non-complete arc-transitive graph of valency $r$ and order $2p$,
where $r$ and $p$ are odd primes.
Then $r\le p$ and one of the following holds.
\begin{itemize}
\item[(1)] $p=5$, $r=3$, $\Sig\cong\O_2$ and $\Aut\Sig\cong\S_5$.
\item[(2)] $\Sig\cong\K_{p,p}$ and $\Aut\Sig\cong\S_p\wr\S_2$.
\item[(3)] $\Sig\cong\CD(2p,r)$ with $r\mid (p-1)$, and one of the following is true.
\begin{itemize}
\item[(i)] $(p,r)=(7,3)$ and $\Aut\Sig\cong\PGL(2,7)$;
\item[(ii)] $(p,r)=(11,5)$ and $\Aut\Sig\cong\PGL(2,11)$;
\item[(iii)] $(p,r)\ne(7,3)$ and $(11,5)$, and $\Aut\Sig\cong\D_{2p}:\ZZ_r$.
\end{itemize}
\end{itemize}
\end{lemma}

A classification of arc-transitive cyclic covers of prime-valent graphs with order
twice a prime is obtained by \cite{PHD}. By \cite[Theorem 1.1]{PHD}
(together with \cite[Theorem 1.1]{Pan-Huang} for a detailed description in one case),
we have the following result.

\begin{theorem}\label{Thm-CycCov}
Let $\Ga$ be an arc-transitive $\ZZ_n$-cover of a graph $\Sig$,
with odd prime valency $r$ and twice a prime order $2p$.
Then one of the following statements holds.
\begin{itemize}
\item[(1)] $\Sig\cong\K_4$, $n=2$ and $\Ga\cong\K_{4,4}-4\K_2$, or $n=4$
and $\Ga\cong \P(8,3)$ is the generalized Petersen graph.

\item[(2)] $\Sig\cong\O_2$, $n=2$, and $\Ga$ is the Dodecahedron graph
or the standard double cover of $\O_2$.

\item[(3)] $\Sig\cong\K_{2p}$ with $p\ge 3$ and $2p-1=r$,
$n=2$ and $\Ga\cong\K_{2p,2p}-2p\K_2$.

\item[(4)] $\Sig\cong\K_{p,p}$ and $n=p^sp_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}$,
where $s\le 1$, $0\le t$ and $p_1,p_2,\dots,p_t$
are distinct primes such that $p\mid (p_i-1)$ for $i=1,2,\dots,t$.

\item[(5)] $\Sig\cong\CD(2p,r)$ with $r\mid (p-1)$,
and $n=r^sp_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}$, where $s\le 1$,
$0\le t$ and $p_1,p_2,\dots,p_t$
are distinct primes such that $r\mid (p_i-1)$ for $i=1,2,\dots,t$.
\end{itemize}
\end{theorem}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Technical lemmas}

In this section, we prove some technical lemmas for the subsequent discussion.

For a group $G$, the socle of $G$, denoted by $\soc(G)$,
is the product of all minimal normal subgroups of $G$.
Obviously, $\soc(G)$ is a characteristic subgroup of $G$.

\begin{lemma}\label{NorCyc-gp}
Suppose $G\cong(\ZZ_{p^s}\times\ZZ_p):\ZZ_3$,
where $p\equiv 1~(\mod 3)$ is a prime
and $s\ge 1$.
Then $G$ has a normal cyclic subgroup of order $p^s$.
\end{lemma}

\begin{proof}
By assumption, we may write $G=(\l a\r\times\l b\r):\l c\r$,
where $o(a)=p^s$, $o(b)=p$ and $o(c)=3$.

If $s=1$, then $G\cong\ZZ_p^2:\ZZ_3$.
Noting that $\ZZ_p^2$ has exactly $p+1$ subgroups isomorphic to
$\ZZ_p$.
Since $3\mid (p-1)$,
$p+1\equiv 2~(\mod 3)$,
hence the action of $\l c\r\cong\ZZ_3$ on these $p+1$ subgroups
is not fixed-point free. So at least one such subgroup must be
invariant under this action,
namely $\l c\r$ normalizes a cyclic subgroup $\ZZ_p$, which is
clearly normal in $G$, the lemma is true in this case.

Assume $s\ge 2$. Let $H=\l a\r\times\l b\r$.
Then $\soc(H)=\l a^{p^{s-1}}\r\times\l b\r\cong\ZZ_p^2$
is a characteristic subgroup of $H$, and so is normal in $G$.
Hence $\l c\r$ normalizes $\soc(H)$.
As $\l c\r\cong\ZZ_3$ acts reducibly on $\soc(H)$ and
$(3,p)=1$,
by Maschke's theorem (see \cite[Theorem 1.4]{Webb}),
$\l c\r$ acts completely reducible on $\soc(H)$,
that is, we may write $\soc(H)=\l b_1,b_2\r$
such that $c$ normalizes both $\l b_1\r$ and $\l b_2\r$.
Noting that at least one of $b_1$ and $b_2$ is not in $\l a\r$,
say $b_1$, then $H=\l a\r\times\l b_1\r$.
Hence, replacing $b$ by $b_1$ if necessary,
we may assume $b^c=b^k$ for some
$k$. As $o(b^k)=o(b)=p$, $(k,p)=1$.

If $a^c\in\l a\r$, the group $\l a\r$ is a required normal cyclic subgroup.
Suppose $a^c=a^ib^j$ with $p{\not |}~j$.
If $p\mid (k-i)$,
then $b^i=b^k$ and $(i,p)=1$,
by a simple computation, we have
$a^{c^2}=(a^ib^j)^c=(a^ib^j)^ib^{kj}=a^{i^2}b^{2ij}$,
and $a^{c^3}=(a^ib^j)^{i^2}b^{2ijk}=a^{i^3}b^{3i^2j}{\not\in}~\l a\r$
as $p{\not |}~3i^2j$,
which is a contradiction because $o(c)=3$.
So $p{\not |}~(k-i)$, and hence the congruence equation
$$(k-i)x\equiv -j~(\mod p)$$
has a solution, say $x=l$.
Now, as $o(ab^l)=p^s$
and $(ab^l)^c=a^ib^jb^{kl}=a^ib^{il}=(ab^l)^i$,
$\l ab^l\r\cong\ZZ_{p^s}$ is normal in $G$.\qed
\end{proof}

The next lemma presents some properties of
the automorphism groups of metacyclic groups isomorphic to $\ZZ_m:\ZZ_2$.

\begin{lemma}\label{m:2}
Let $K=\l a\r:\l b\r\cong\ZZ_m:\ZZ_2$ be a metacyclic group,
where $m=2^dn>1$ with $d\ge 0$ and $n$ odd.
Then the following statements hold.
\begin{itemize}
\item[(1)] If $m$ is odd, then $K\cong\ZZ_s\times\D_{2t}$, where $st=m$, and $(s,t)=1$.
\item[(2)] If $d=1$, then either $K$ is abelian,
or the center $Z(K)\cong\ZZ_{2n_1}$ and $K/Z(K)\cong\D_{2n_2}$,
where $n_1n_2=n$, $n_2>1$ is odd and $(n_1,n_2)=1$.
\item[(3)] If $d\ge 1$, then $\l a^2\r$ is a characteristic subgroup of $K$ with index $4$.
\item[(4)] Suppose $d\ge 2$ and $K$ is normal in an overgroup $X$.
Then $X$ has a normal subgroup which is contained in $K$ with index $2$.
\end{itemize}
\end{lemma}

\begin{proof}
(1) Suppose that $\l a^e\r\cong\ZZ_{p^l}$ is a Sylow $p$-subgroup of $\l a\r$,
where $p$ is an odd prime dividing $m$
and $l$ is a positive integer.
Then $\l a^e\r\lhd K$, so $(a^e)^b=(a^{e})^k$ for some integer $k$.
Since $b^2=1$, $k^2\equiv 1~(\mod p^{l})$, and so $p^l\mid (k-1)(k+1)$.
As $(k-1,k+1)$ divides $2$ and $p^{l}\ge 3$, we have
$k\equiv\pm 1~(\mod p^l)$.
Hence either $b$ centralizes $\l a^e\r$, or $\l a^e,b\r\cong\D_{2p^l}$.
Now, as $\l a\r\cong\ZZ_{p_1^{s_1}}\times\cdots\times\ZZ_{p_k^{s_k}}$
for some distinct odd primes $p_1,\dots,p_k$,
and $\l b\r\cong\ZZ_2$ acts trivially or dihedrally
on each Sylow $p$-subgroup of $\l a\r$,
collecting factors on which $\l b\r$
acts trivially gives the center $\ZZ_s$ of $K$
while collecting the remaining factors together with $\l b\r$ gives
the dihedral group $\D_{2t}$.
Hence $K\cong\ZZ_s\times\D_{2t}$.

(2) Since $d=1$, $\l a^n\r\in Z(K)$
and $K=\l a^n\r\times(\l a^2\r:\l b\r)$.
By part (1), $\l a^2\r:\l b\r\cong\ZZ_{n_1}\times\D_{2n_2}$,
where $n_1n_2=n$ and $(n_1,n_2)=1$.
If $n_2=1$, $K$ is abelian;
if $n_2>1$, then $Z(K)\cong\ZZ_{2n_1}$ and $K/Z(K)\cong\D_{2n_2}$.

(3) Suppose $a^b=a^q$ with $q$ an integer. Then
$q^2\equiv 1~(\mod m)$, so $q$ is odd as $m$ is even.
Let $K^{(2)}=\l g^2\mid g\in K\r$.
Clearly, $K^{(2)}\supseteq\l a^2\r$ is a characteristic subgroup of $K$.
Since each element of $K$ has a form $a^i$ or $a^ib$ for some integer $i$,
$(a^i)^2\in\l a^2\r$ and $(a^ib)^2=a^{(1+q)i}\in\l a^2\r$,
we conclude that $\l a^2\r=K^{(2)}$ is a characteristic subgroup of $K$ with index $4$.

(4) By part (3), $\l a^2\r$ is a characteristic subgroup of $K$,
so is $\l a^4\r$, hence $\l a^4\r\lhd X$ and $K/\l a^4\r\lhd X/\l a^4\r$.
Since $K/\l a^4\r\cong\ZZ_4:\ZZ_2$ is a split extension,
we have $K/\l a^4\r\cong\ZZ_4\times\ZZ_2$ or $\D_8$.
For the former case, $K/\l a^4\r$ has a characteristic subgroup,
say $M/\l a^4\r$, such that $M/\l a^4\r\cong\ZZ_2^2$,
so $M/\l a^4\r\lhd X/\l a^4\r$,
it follows that $M$ is normal in $X$ and is of index $2$ in $K$.
For the latter case, $K/\l a^4\r$ has a characteristic subgroup,
say $N/\l a^4\r$, such that $N/\l a^4\r\cong\ZZ_4$,
it follows that $N$ is normal in $X$ and  is of index $2$ in $K$.\qed
\end{proof}

The following lemma is regarding normal arc-transitive Cayley graphs.

\begin{lemma}\label{Cov-val(r)}
Let $G\cong\ZZ_2^2\times\D_{2p}$ with $p\ge 5$ a prime.
Then there is no normal arc-transitive Cayley graph of $G$ with prime valency $r\ge 5$.
\end{lemma}

\begin{proof}
Suppose that, on the contrary,
$\Ga=\Cay(G,S)$ is an $X$-normal arc-transitive Cayley graph of $G$,
where $G\lhd~X\le\Aut\Ga$ and $S\subseteq G\setminus\{1\}$.
Set $G=(\l a\r\times\l b\r)\times(\l c\r:\l d\r)$,
where $\l a\r\times\l b\r\cong\ZZ_2^2$
and $\l c\r:\l d\r\cong\D_{2p}$.
By Lemma~\ref{Nor-CGs},
$S=s^{\l\s\r}$,  where $s\in G$ is an involution
 and $\s\in\Aut(G)$ is of order $r$.
Noting that $\l a,b\r\cong\ZZ_2^2$ is a characteristic subgroup of $G$,
the restriction $\s|_{\l a,b\r}$ of $\s$ on $\l a,b\r$ is an automorphism of $\l a,b\r$.
Clearly, the order of $\s|_{\l a,b\r}$ divides $o(\s)=r$, and so is $1$ or $r$.
As $r\ge 5$ and $\Aut(\ZZ_2^2)\cong\S_3$
has no element with order bigger than $3$, we further conclude
that $\s|_{\l a,b\r}$ is the identity automorphism of $\l a,b\r$.
If $s\in\l a,b\r$, then
$\l S\r=\l s^{\l\s\r}\r\subseteq\l a,b\r<G$,
$\Ga$ is disconnected, which is a contradiction.
Thus $s\in G\setminus\l a,b\r$,
and so $s=a^ib^jc^kd$ for some integers $i,j,k$.
Noting that there is an automorphism $\t$ which
fixes $a,b,c$ and maps $s$ to $d$,
and $\Cay(G,S)\cong\Cay(G,S^{\t})$,
we may, up to isomorphism, assume $s=d$.
Now, since $\l c\r\cong\ZZ_p$ is a characteristic subgroup of $G$,
$\s$ has a form:
$$\s:a\to a,~b\to b,~c\to c^k,~d\to a^ub^vc^wd.$$
By computation, $d=d^{\s^r}=a^{ru}b^{rv}c^{w(1+k+\cdots+k^{r-1})}d$.
So $a^{ru}=b^{rv}=1$, implying $a^u=b^v=1$ as $r$ is odd
and $a,b$ are involutions.
It follows that $\l s^{\l\s\r}\r=\l d^{\l\s\r}\r\subseteq\l c,d\r<G$,
which contradicts the connectivity of $\Ga$.\qed
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Metacyclic covers of $K_{p,p}$}

For convenience, we give the following Hypothesis 4.1
and Notations 4.2, which will be used
throughout the rest of the paper.

\vskip0.1in
{\noindent\bf Hypothesis $4.1.$}
Let $\Ga$ be an $X$-arc-transitive $K$-cover of a graph $\Sig$,
where $X\le\Aut\Ga$,
$\Sig$ is a non-complete graph of odd prime valency $r$ and order $2p$
with $p$ a prime,
and $K:=\l a\r:\l b\r\cong\ZZ_m:\ZZ_q$ is
a metacyclic but not cyclic group with $q<r$ a prime.
Then $K\lhd X$ is semiregular on $V\Ga$,
and $Y:=X/K\le\Aut\Sig$ acts transitively on the arc set of $\Sig$.
\vskip0.1in

{\noindent\bf Notation $4.2.$}
If $\Sig$ is a bipartite graph,
let $\Del_1$ and $\Del_2$ be the biparts of $\Sig$,
let $Y^+=Y_{\Del_1}=Y_{\Del_2}$ be the stabilizer of $Y$ on the biparts,
and let $X^+=K.Y^+$ (the full preimages of $Y^+$ under the natural homomorphism
of $Y$ to $Y/K$).
Then $|X:X^+|=|Y:Y^+|=2$ and
$Y=\l Y^+,y\r$ for some $y\in Y\setminus Y^+$.
\vskip0.1in

By Hypothesis 4.1, the tuple $(\Sig,\Aut\Sig)$ is listed in Lemma~\ref{2p-graphs}.
We will prove Theorem~\ref{Thm-1} by analysing all the candidates $\Sig$
in Lemma~\ref{2p-graphs}.
Recall that, for a group $G$, a subgroup
$H$ is called the {\it Frattini subgroup} of $G$
if $H$ is the intersection of all maximal subgroups of $G$;
and for a prime $t\mid |G|$,
$H$ is called a {\it Hall $t'$-subgroup} if $(|H|,t)=1$ and $|G:H|$ is a $t$-power.
It is well known that the Frattini subgroup and normal Hall $t'$-subgroups
are characteristic subgroups.
As usual, we use $G_t$ and $G_{t'}$ to denote a Sylow $t$-subgroup
and a Hall $t'$-subgroup of $G$, respectively.

First, we prove an induction lemma.


\begin{lemma}\label{Induction}
With notation above, the following statements hold.
\begin{itemize}
\item[(1)] If $q{\not |}~m$, then $q=2$ and $\Sig\cong\O_2$.
\item[(2)] If $q\mid m$, then there exists a graph $\Ome$
being an arc-transitive $\ZZ_q^2$-cover of $\Sig$.
\end{itemize}
\end{lemma}

\begin{proof}
If $q{\not |}~m$, then $\l a\r$ is a characteristic subgroup of $K$,
so is normal in $X$.
Since $K/\l a\r\cong\ZZ_q$,
by Theorem~\ref{Praeger}(4),
the normal quotient graph $\Ga_{\l a\r}$
is an $X/\l a\r$-arc-transitive
$\ZZ_q$-cover of $\Sig$.
Then, as $q<r=\val(\Sig)$ and $\Ga$ is not a complete graph,
by Theorem~\ref{Thm-CycCov},
we have $q=2$ and $\Sig\cong\O_2$, as in part (1) of the lemma.

Assume now $q\mid m$.
Then $m=q^dn$ with $d\ge 1$ and $(q,n)=1$.
Clearly, $\l a^{q^d}\r\cong\ZZ_n$ is a normal Hall $q'$-subgroup of $K$,
so is characteristic in $K$ and normal in $X$.
Let $\Ga_1$ denote the normal quotient graph $\Ga_{\l a^{q^d}\r}$.
By Theorem~\ref{Praeger}, $\Ga_1$ is an $X/\l a^{q^d}\r$-arc-transitive
 $K/\l a^{q^d}\r$-cover of $\Sig$.
 Let $\Phi$ be the Frattini subgroup of $K/\l a^{q^d}\r$.
Then $\Phi$ is characteristic in $K/\l a^{q^d}\r$ and normal in $X/\l a^{q^d}\r$.
Since $K/\l a^{q^d}\r\cong\ZZ_{q^d}:\ZZ_q$,
by \cite[5.3.2]{Robinson}, $(K/\l a^{q^d}\r)/\Phi\cong\ZZ_q^2$.
It then follows from Theorem~\ref{Praeger} that $\Ome:=(\Ga_1)_{\Phi}$
is an arc-transitive $\ZZ_q^2$-cover of $\Sig$.\qed
\end{proof}

\vskip0.2in
In this section, we will determine all arc-transitive $K$-covers of
the complete bipartite graph $\K_{p,p}$.

The following lemma excludes the case where $p\ge 5$.

\begin{lemma}\label{K(p,p)}
Suppose $\Sig\cong\K_{p,p}$ with $p\ge 5$. Then no graph $\Ga$ exists.
\end{lemma}

\begin{proof}
By Lemma~\ref{Induction},
it is sufficient to show that
Lemma~\ref{K(p,p)} is true for the case where $K\cong\ZZ_q^2$.
Recall that $q<r=\val(\Sig)=p$.

Let $K_1$ and $K_2$ be the kernels of $Y^+$ acting on $\Del_1$ and $\Del_2$, respectively.
For any $\a\in\Del_1$, since $y{\not\in}~Y^+$,
we have $\a^{y^{-1}}\in\Del_2$
and $\a^{y^{-1}K_2y}=(\a^{y^{-1}})^{K_2y}=(\a^{y^{-1}})^y=\a$,
so $K_2^y=y^{-1}K_2y\subseteq K_1$.
Similarly, $K_1^y\subseteq K_2$.
Thus $K_1=K_2^y$ and
$K_2=K_1^y$.
Since $Y$ is arc-transitive on $\Sig$,
$Y_{\a}$ is transitive on the neighbor set $\Sig(\a)=\Del_2$.
If $Y^+$ acts faithfully on $\Del_1$,
then $Y^+$ can be viewed as a transitive permutation group of degree $p$ on $\Del_1$,
so $Y^+\le\S_p$.
It follows that $Y^+_{\a}\le\S_{p-1}$,
which contradicts that $Y^+_{\a}=Y_{\a}$ is transitive on $\Del_2$ of size $p$.

Hence, $Y^+$ acts unfaithfully on both $\Del_1$ and $\Del_2$,
that is, $K_1\cong K_2\ne 1$.
Clearly, $K_1\cap K_2=1$,
so $K_1K_2=K_1\times K_2\lhd Y^+$
and $Y^+\cong (K_1\times K_2).P$,
where $P\cong Y^+/(K_1\times K_2)$.
It follows that $K_i.P\cong Y^+/K_{3-i}=(Y^+)^{\Del_{3-i}}$
is a transitive permutation group of degree $p$,
where $i=1,2$.
By the classification of transitive permutation groups of prime degree (see
\cite[P. 99]{Dixon}), we have that either $K_i.P\le\ZZ_p:\ZZ_{p-1}$ is an affine group
or $K_i.P$ is
an almost simple $2$-transitive group.
Let $T_i=\soc(K_i.P)$  and let $G=K.(T_1\times T_2)$.

If $T_i$ is nonabelian, then $T_i$ is 2-transitive on $\Del_i$,
so $T_1\times T_2$ is transitive
on the set of all $4$-cycles of $\Sig\cong\K_{p,p}$.
Noting that $G\cong\ZZ_q^2.(T_1\times T_2)$ is a central extension
by Lemma~\ref{cent-ext},
then with quite similar discussion as in \cite[Theorem 4.2]{DMM}
(or \cite[Lemma 5.1]{Pan-Huang}),
we have that $\Ga$ is disconnected,
which is a contradiction.

Assume now that $T_i$ is abelian. Then $T_1\times T_2\cong\ZZ_p^2$ is
edge-transitive and vertex-intransitive on $\Sig$,
and so $G=K.(T_1\times T_2)\cong\ZZ_q^2:\ZZ_p^2$ is
edge-transitive and vertex-intransitive on $\Ga$.
By Lemma~\ref{GLP}, $\Ga\cong\Cos(G,G_{\a},G_{\b})$
for some adjacent vertices $\a$ and $\b$ of $\Ga$.
Clearly, $G_{\a}\cong G_{\b}\cong\ZZ_p$.
If $p\ge q+2$, then $|\Aut(\ZZ_q^2)|=|\GL(2,q)|=q(q-1)^2(q+1)$
is not a multiple of $p$, by Lemma~\ref{N/C},
the centralizer $C_G(K)=G$,
and so $G\cong\ZZ_q^2\times\ZZ_p^2$ is abelian.
It follows that $\l G_{\a},G_{\b}\r\le G_p<G$, so
$\Ga$ is disconnected by Lemma~\ref{GLP}, which is a contradiction.
Thus $p\le q+1$ and hence $q=p-1$ as $q<p$.
Now, both $p$ and $p-1$ are primes, the only possibility is
$p=3$ and $q=2$, which is a contradicts to the assumption $p\ge 5$.\qed
\end{proof}

We next consider the case $\Sig\cong\K_{3,3}$.

\begin{lemma}\label{K(3,3)}
Suppose $\Sig\cong\K_{3,3}$.
Then $K\cong\ZZ_{2n}\times\ZZ_2$ with $n$ odd,
and $\Ga$ is characterized in \cite[Theorem 5.1]{Conder-1}.
\end{lemma}

\begin{proof}
By assumption, $r=\val(\Sig)=3$,
so $q=2$ and $K=\l a\r:\l b\r\cong\ZZ_m:\ZZ_2$.
Then Lemma~\ref{Induction}(1) implies
that $m$ is even.
Set $m=2^dn$ with $d\ge 1$ and $n$ odd.

If $d\ge 2$, since $K\lhd X$, by Lemma~\ref{m:2}(4),
$X$ has a normal subgroup, say $M$,
such that $M$ is contained in $K$ with index $2$.
Then, by Theorem~\ref{Praeger}, $\Ga_{M}$ is an arc-transitive $\ZZ_2$-cover of $\Sig$,
so $|V\Ga_M|=12$; however,
there is no cubic arc-transitive graph of order $12$ by \cite{Cub-768},
which is a contradiction.

Therefore, $d=1$.
Assume $K$ is nonabelian.
Then $\l a^{n}\r\cong\ZZ_2$ is in the center of $K$,
$\l a^2\r$ is a characteristic subgroup of $K$
and as $K$ is nonabelian, $b$ is not in the center of $K$,
we conclude that $\l a\r=\l a^2\r\times\l a^n\r$ is normal in $X$.
By Theorem~\ref{Praeger}, $\Ga_{\l a\r}$ is an arc-transitive $\ZZ_2$-cover of $\Sig$,
in particular, $|V\Ga|=12$,
which is a contradiction by \cite{Cub-768}.
Thus $K$ is abelian, $K\cong\ZZ_{2n}\times\ZZ_2$ with $n$ odd,
and $\Ga$ is characterized in \cite[Theorem 5.1]{Conder-1}.\qed
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Metacyclic covers of $\CD(2p,r)$}

Let $\Ga,X,\Sig,K$ and $Y$ be as in Hypothesis 4.1,
and as $\CD(2p,r)$ is a bipartite graph, we use Notation 4.2.

In this section, we will determine all arc-transitive $K$-covers of
the bipartite graphs $\CD(2p,r)$ with $r\mid (p-1)$.
Noting that $K=\l a\r:\l b\r\cong\ZZ_m:\ZZ_q$ and $q<r$ are primes.

\begin{lemma}\label{(2p,r)}
Assume $\Sig\cong\CD(2p,r)$ with $r\mid (p-1)$ and $Y\cong\D_{2p}:\ZZ_r$.
Then $q=2$ and $r=3$.
\end{lemma}

\begin{proof}
Suppose that, on the contrary, $q\ge 3$.
Since $Y\cong\D_{2p}:\ZZ_r$,
$\Sig$ is a $Y$-normal arc-transitive Cayley graph of the dihedral group $\D_{2p}$,
so $\Ga$ is an $X$-normal arc-transitive Cayley graph of $R:=K.\D_{2p}$.
Also, by Lemma~\ref{Induction}(1), we have $q\mid m$.

We first consider the case where $m=q$.
Then $R\cong\ZZ_q^2:\D_{2p}$
and $X=K.Y\cong R:\ZZ_r$.
Since $K$ is abelian, the centralizer $C:=C_R(K)\supseteq K$.
If $C=K$, Lemma~\ref{N/C} implies
$\D_{2p}\cong R/K=R/C\le\Aut(K)\cong\Aut(\ZZ_q^2)\cong\GL(2,q)$,
as $3\le q<r<p$, $|\GL(2,q)|=q(q-1)^2(q+1)$ is not a multiple of $p$,
which is a contradiction.
On the other hand, if $C=R$, then $R=K\times R_{q'}$,
so $R_{q'}\cong\D_{2p}$ is normal in $X$ and has $q^2$ orbits on $V\Ga$,
by Theorem~\ref{Praeger},
$\Ga_{R_{q'}}$ is arc-transitive of
odd valency $r$ and odd order $q^2$, which is also a contradiction.
Hence $K<C<R$, and $C/K\ne 1$ is a proper normal subgroup of $R/K\cong\D_{2p}$,
we further conclude that $C=R_{2'}\cong\ZZ_q^2\times\ZZ_p$
and $\ZZ_p\cong R_p\lhd X$.
Now, Theorem~\ref{Praeger} implies that $\Ga_{R_p}$ is an $X/R_p$-normal
arc-transitive Cayley graph
of the group $R/R_p$.
Clearly, $R/R_p\cong R_{p'}=K:\ZZ_2\cong \ZZ_q^2:\ZZ_2$.
As $q\ge 3$, $R_{p'}$ is nonabelian by Lemma~\ref{Nor-CGs}.
Suppose that $c\in R_{p'}$ is an involution.
If $x^c=y~{\not\in}\l x\r$ for some $x\in K$,
as $o(c)=2$, $y^c=x$,
so $(xy)^c=yx=xy$ and $(xy^{-1})^c=yx^{-1}=(xy^{-1})^{-1}$,
hence $R_{p'}=\l xy\r\times\l xy^{-1},c\r\cong\ZZ_q\times\D_{2q}$.
Since $\Ga_{R_p}$ is an $X/R_p$-normal Cayley graph with valency $r$ of
$R/R_p$, we may assume $\Ga_{R_p}=\Cay(R/R_q, S)$,
where $S\subseteq R/R_p\setminus\{1\}$.
By Lemma~\ref{Nor-CGs}, elements in $S$ are involutions,
and as $R/R_p\cong R_{p'}\cong\ZZ_q\times\D_{2q}$,
we have $\l S\r\le\D_{2q}<R/R_p$,
$\Ga$ is disconnected, which is a contradiction.
Therefore, $x^{c}\in\l x\r$ for each $x\in K$,
and $\l x,c\r\cong\ZZ_{2q}$ or $\D_{2q}$ by Lemma~\ref{m:2}(1).
Since $R_{p'}$ is nonabelian, and as proved above, $R_{p'}{\not\cong}\ZZ_{q}\times\D_{2q}$,
we further have that
$R_{p'}\cong\Dih(\ZZ_q^2)$ is a generalized dihedral group.
Now, as $X/R_p\cong(R/R_p):\ZZ_r\cong\ZZ_q^2:\ZZ_r:\ZZ_2$,
and $(X/R_p)_{u}\cong\ZZ_r$ with $u\in V(\Ga_{R_p})$ is not normal in $X/R_p$,
by Lemma~\ref{N/C}, $\ZZ_q^2$ in self-centralized in $X/R_p$
and $\ZZ_r:\ZZ_2\le \Aut(\ZZ_q^2)\cong\GL(2,q)$.
Hence $r$ divides $|\GL(2,q)|=q(q-1)^2(q+1)$.
However, as  $q<r$ are primes, the only possibility is $q=2$ and $r=3$,
which contradicts the assumption that $q\ge 3$.

For the general case, by Lemma~\ref{Induction},
there is a graph $\Ome$ which is
an arc-transitive $\ZZ_q^2$-cover of $\Sig$,
then above discussion also leads to a contradiction.

Therefore, $q=2$ and $R\cong\ZZ_2^2.\D_{2p}$.
We now prove $r=3$.
Since $r\ge 3$ divides $p-1$,
$p\ge 7$,
and as $|\Aut(\ZZ_2^2)|=|\S_3|=6$,
Lemma~\ref{N/C} implies $R=(\ZZ_2^2\times\ZZ_p).\ZZ_2$
and so $R_p\lhd X$.
By Theorem~\ref{Praeger},
$\Ga_{R_p}$ is an $X/R_p$-normal arc-transitive Cayley graph
of the group $R/R_p\cong R_2\cong\ZZ_2^2.\ZZ_2$.
If $R_2$ is nonabelian, then $R_2\cong Q_8$
is a quaternion group or isomorphic to $\D_8$.
The former case is not possible by Lemma~\ref{Nor-CGs} as $Q_8$ has unique involution,
and the latter case is not possible by \cite[Lemma 3.1]{Pan}.
Thus $R_2$ is abelian.
By Lemma~\ref{Nor-CGs}, $R_2\cong\ZZ_2^3$
and $R\cong\ZZ_2^2\times\D_{2p}$.
Recall that $\Ga$ is an $X$-normal arc-transitive Cayley graph of $R$,
by Lemma~\ref{Cov-val(r)}, we have $r<5$,
hence $r=3$, as required.\qed
\end{proof}

The next lemma excludes the case where $\Sig\cong\CD(22,5)$.

\begin{lemma}\label{(22,5)}
Assume $\Sig\cong\CD(22,5)$. Then no graph $\Ga$ exists.
\end{lemma}

\begin{proof}
With Lemma~\ref{Induction},
we only need to prove the lemma
for the case $K\cong\ZZ_q^2$.

Since $\Sig\cong\CD(22,5)$,
$\Aut\Sig\cong\PGL(2,11)$ by Lemma~\ref{2p-graphs}.
Since $Y\le\Aut\Sig$ acts arc-transitively on $\Sig$,
$110$ divides $|Y|$,
by Atlas \cite{Atlas},
$Y\cong\D_{22}:\ZZ_5,\PSL(2,11)$ or $\PGL(2,11)$.
Noting that $\CD(22,5)$ is a bipartite graph,
$Y$ has a normal subgroup $Y^+$ with index 2,
so $Y{\not\cong}~\PSL(2,11)$. If $Y{\cong}~\D_{22}:\ZZ_5$,
by Lemma~\ref{(2p,r)}, we have $\val(\Sig)=3$, yielding a contradiction.

Assume $Y\cong\PGL(2,11)$. Then $Y^+\cong\PSL(2,11)$
and $Y_{\d}=Y^+_{\d}\cong\A_5$ as $|V\Sig|=22$, where $\d\in V\Sig$.
By Lemma~\ref{cent-ext}, $X^+=K.Y^+\cong \ZZ_q^2.\PSL(2.11)$ is a central extension,
$X^+=K(X^+)'$, and as the Schur multiplier of $\PSL(2,11)$ is $\ZZ_2$
(see \cite[P. 302]{Gorenstein}),
$(X^+)'\cong\PSL(2,11)$ or $\SL(2,11)$.
For $\a\in V\Ga$,
by Theorem~\ref{Praeger}(3), $(X^+)'_{\a}\lhd X^+_{\a}=X_{\a}\cong Y_{\d}\cong\A_5$,
so $(X^+)'_{\a}=1$ or $\A_5$.
If $(X^+)'_{\a}=1$, as $X^+=K(X^+)'$,
 Lemma~\ref{Stabilizer} implies
$\A_5\cong X^+_{\a}\le K/(K\cap (X^+)')$, which is a contradiction as $K$ is abelian.
Thus $(X^+)'_{\a}\cong\A_5$,
and hence $(X^+)'~{\not\cong}~\SL(2,11)$ since $\SL(2,s)$ with $s$ an odd prime power has a unique involution
by \cite[Lemma 7.4]{Isaacs}.
Therefore, $(X^+)'\cong\PSL(2,11)$, $X^+=K\times (X^+)'$ and each orbit of $(X^+)'$ in $V\Ga$
has length $|\PSL(2,11):\A_5|=11$.
Since $|V\Ga|=22q^2$, $(X^+)'\lhd X$ has $2q^2\ge 8$ orbits on $V\Ga$.
By Theorem~\ref{Praeger}, $\Ga_{(X^+)'}$ is an $X^+/(X^+)'$-arc-transitive graph, which is
also a contradiction as $X^+/(X^+)'\cong\ZZ_q^2$ is abelian.\qed
\end{proof}

With Lemmas~\ref{(2p,r)} and \ref{(22,5)},
we may now determine all
arc-transitive $K$-covers of $\CD(2p,r)\!$ with $r\mid (p-1)$.

\begin{lemma}\label{CF-(2p,r)}
Assume $\Sig\cong\CD(2p,r)$ with $r\mid (p-1)$.
Then $r=3$ and $\Ga$ is as in part $(2)$ of Theorem~$\ref{Thm-1}$.
\end{lemma}

\begin{proof}
By Lemma~\ref{(22,5)}, $(p,r)\ne (11,5)$.
If $(p,r)=(7,3)$, the $q=2$ as $q<r$;
if $(p,r)\ne (11,5)$ and $(7,3)$,
by Lemma~\ref{2p-graphs},
$\Aut\Sig\cong\D_{2p}:\ZZ_r$ is arc-regular on $\Sig$,
so $Y=\Aut\Sig$ as $Y$ is arc-transitive on $\Sig$,
by Lemma~\ref{(2p,r)}, we also have $q=2$ and $r=3$.
Now, by Lemma~\ref{Induction}, $m$ is even.
Suppose $m=2^dn$ with $(2,n)=1$ and $d\ge 1$.

Assume $d\ge 2$.
By Lemma~\ref{m:2}(4),
$X$ has a normal subgroup, say $M$,
such that $M$ is contained in $K$ with index $2$.
It then follows from Theorem~\ref{Praeger}
that $\Ga_{M}$ is an arc-transitive $\ZZ_2$-cover of $\CD(2p,3)$,
which is impossible
by Theorem~\ref{Thm-CycCov}(5).

Hence, $d=1$ and $K=\l a\r:\l b\r\cong\ZZ_{2n}:\ZZ_2$.
If $n=1$, by \cite[Theorem 3.1]{Zhou-Feng},
$\Ga\cong\CGD(8p,3)$ (as $CD_{8p}$ there),
part (2)(i) of Theorem~\ref{Thm-1} holds.
If $K$ is nonabelian,
by Lemma~\ref{m:2}(2), the center $Z(K)\cong\ZZ_{2n_1}$ and $K/Z(K)\cong\D_{2n_2}$,
where $n_1n_2=n$, $n_2>1$ and $(n_1,n_2)=1$.
Then, by Theorem~\ref{Praeger},
$\Ga_{Z(K)}$ is an arc-transitive $\D_{2n_2}$-cover of $\Sig$,
as $n_2$ is odd, which is not possible by \cite[Theorem 3.1]{Zhou-Feng}.
Hence $K\cong\ZZ_{2n}\times\ZZ_2$ is abelian.
Since $\l a^2\r\lhd X$
and $K/\l a^2\r\cong\ZZ_2^2$, by Theorem~\ref{Praeger},
$\Ga_{\l a^2\r}$ is an $X/\l a^2\r$-arc-transitive
$\ZZ_2^2$-cover of $\Sig\cong\CD(2p,3)$,
then \cite[Theorem 3.1]{Zhou-Feng} implies
$\Ga_{\l a^2\r}\cong\CGD(8p,3)$
is a $1$-arc-regular normal Cayley graph
of the generalized dihedral group $\Dih(\ZZ_{2p}\times\ZZ_2)$,
so $X/\l a^2\r\cong\Dih(\ZZ_{2p}\times\ZZ_2):\ZZ_3$.
Noting that $\l a^2\r.\ZZ_2^2\cong K\lhd X$,
we further conclude $X\cong(\ZZ_{2n}\times\ZZ_2).\D_{2p}.\ZZ_3$.
Further, as $K_2\cong\ZZ_2^2$ is normal in $X$
and $K/K_2\cong\ZZ_n$, by Theorem~\ref{Praeger},
$\Ga_{K_2}$ is an arc-transitive $\ZZ_{n}$-cover of $\CD(2p,3)$.
It then follows from Theorem~\ref{Thm-CycCov}(5) that
$n=3^sp_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}$, where $s\le 1$,
$0\le t$, and $p_1,p_2,\dots,p_t$ are distinct primes
such that $3\mid p_i-1$ for $i=1,2,\dots,t$.

Recall that $X\cong(\ZZ_{2n}\times\ZZ_2).\D_{2p}.\ZZ_3$.
Let $R\lhd X$ such that $R\cong K.\D_{2p}$,
and let $C=C_R(K_{2'})$.
Then $K\subseteq C\lhd R$.
If $C=K$,  Lemma~\ref{N/C} implies
$\D_{2p}\cong Y/K=Y/C\le\Aut(K_{2'})\cong \ZZ_{n}^{*}$,
yielding a contradiction.
Thus $C>K$ and $1\ne C/K\lhd Y/K\cong\D_{2p}$,
implying $C\supseteq K.\ZZ_p$.
Let $L\subseteq C$ such that $L/K\cong\ZZ_p$.
Then $L$ is abelian and $L\cong \ZZ_{2np}\times\ZZ_2$,
or $\ZZ_{2n}\times\ZZ_{2p}$ with $p\mid n$.
Clearly, $L\subseteq K.\D_{2p}$ is semiregular and has exactly two orbits on $V\Ga$,
by \cite[Lemma 3.1]{Zhou-Feng}, $\Ga$ is a Cayley graph of the generalized dihedral group
$\Dih(L)$.
Then, by \cite[Proposition 2.6]{Zhou-Feng}, we have
$\Ga\cong\CGD(2,np,\lambda)$, or $\CGD(2p,n/p,\lambda)$ with $p\mid n$,
as in part (2)(ii) of Theorem~\ref{Thm-1}.\qed
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem~\ref{Thm-1}}

To complete the proof of Theorem~\ref{Thm-1},
we first prove that the graphs $\Ga$ in part $(2)$(ii) of Theorem~\ref{Thm-1}
are really examples.

\begin{lemma}\label{K-cov}
Suppose that $p\equiv 1~(\mod 3)$ is a prime
and $m=2\cdot 3^sp_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}$, where $s\le 1$, $0\le t$,
and $p_1,p_2,\dots,p_t$ are distinct primes such that
$3\mid (p_i-1)$ for $i=1,2,\dots,t$.
Then the graphs $\CGD(2,{mp\over 2},\lambda)$
and $\CGD(2p,{m\over 2p},\lambda)$ with $p\mid m$
are arc-transitive $(\ZZ_{m}\times\ZZ_2)$-covers of $\CD(2p,3)$.
\end{lemma}

\begin{proof}
(i) Let $\Omega=\CGD(2,{mp\over 2},\lambda)$.
By Example~\ref{Cay-GenDih}, $\Omega=\Cay(G,\{ a,b,ab^{-\lambda}c\})$,
where
$$G=\l a,b,c\mid a^2=b^{mp}=c^2=1, b^a=b^{-1},c^a=c^{-1},bc=cb\r,$$
and $\lambda^2+\lambda+1\equiv 0~(\mod {mp\over 2})$.
By \cite[Proposition 2.6(4)]{Zhou-Feng}, $\Omega$ is an arc-regular normal Cayley graph of $G$,
hence $\Aut(\Omega)\cong G:\ZZ_3$.

Clearly, the center $Z(G)=\l b^{mp\over 2}\r\times\l c\r\cong\ZZ_2^2$
and $G^{(2p)}:=\l g^{2p}\mid g\in G\r=\l b^{2p}\r\cong\ZZ_{m\over 2}$
are characteristic subgroups of $G$.
Set $M=Z(G)G^{(2p)}$.
Since $({mp\over 2},2p)=(3^sp_1^{r_1}p_2^{r_2}\cdots p_t^{r_t}p,$ $2p)=p$,
we have $M=\l b^{mp\over 2},b^{2p}\r\times\l c\r=\l b^p\r\times\l c\r\cong\ZZ_{m}\times\ZZ_2$
is a characteristic subgroup of $G$,
and hence $M\lhd \Aut(\Omega)$.
By Theorem~\ref{Praeger}, $\Omega$ is an arc-transitive $(\ZZ_{m}\times\ZZ_2)$-cover of
$\Ga_M$, and as $|V(\Omega_M)|=2p$ with $p\equiv 1~(\mod 3)$ and $\val(\Omega_M)=3$,
by Lemma~\ref{2p-graphs}, $\Omega_M\cong\CD(2p,3)$.

(ii) Set $\Omega'=\CGD(2p,{m\over 2p},\lambda)$ with $p\mid m$.
Then $\Omega'=\Cay(G,\{ a,b,ab^{-\lambda}c\})$
by Example~\ref{Cay-GenDih},
where $$G=\l a,b,c\mid a^2=b^{m}=c^{2p}=1, b^a=b^{-1},c^a=c^{-1},bc=cb\r,$$
and $\lambda=0$ if $m=2p$,
and $\lambda^2+\lambda+1\equiv 0~(\mod {m\over 2p})$ if ${m\over 2p}>1$.
By \cite[Proposition 2.6]{Zhou-Feng},
$\Omega'$ is a normal Cayley graph of $G$,
and $\Omega'$ is
$2$-arc-regular if ${m\over 2p}=1$ of $3$,
or is $1$-arc-regular if ${m\over 2p}>3$.
Hence $\Aut(\Omega')\cong\G:\ZZ_3$ or $G:\S_3$,
it follows that there exists an automorphism group $X\le\Aut(\Omega')$
such that $X\cong G:\ZZ_3$ acts arc-regularly on $\Omega'$.
Let $\l f\r=X_{\a}\cong\ZZ_3$ with $\a\in V\Omega'$.
Noting that $Z(G)=\l b^{m\over 2}\r\times\l c^p\r\cong\ZZ_2^2$
and $G_{2'}=\l b^2\r\times\l c^2\r\cong\ZZ_{m\over 2}\times\ZZ_p$
are characteristic subgroups of $G$,
we have $\l b,c\r=\l b^{m\over 2},b^2\r\times\l c^p,c^2\r=Z(G)G_{2'}$
is characteristic in $G$ and so normal in $X$.
Let $H=\l b,c,f\r$. Then $H=\l b,c\r:\l f\r\cong(\ZZ_{m}\times\ZZ_{2p}):\ZZ_3$
and $X=\l H\r:\l a\r$.

\vskip0.1in
{\noindent \bf Claim.} $X$ has a normal subgroup $N\subseteq G$
such that $N\cong\ZZ_{m}\times\ZZ_2$.

\vskip0.1in
Since $p\equiv 1~(\mod 3)$
and $p\mid m$, we may suppose $p=p_1$ and $s_1\ge 1$.
Now, $\l b,c\r_{p'}\cong\ZZ_{m\over p^{s_1}}\times\ZZ_2$ is normal in $H$,
with order ${2m\over p^{s_1}}$.
Consider the group $R:=(\l b^{m\over p^{s_1}}\r\times\l c^2\r):\l f\r$.
Then $R\cong(\ZZ_{p^{s_1}}\times\ZZ_p):\ZZ_3$.
As $3\mid (p-1)$, by Lemma~\ref{NorCyc-gp},
$R$ has a normal cyclic subgroup $M\subseteq\l b,c\r$ with order ${p^{s_1}}$.
Noting that $M$ is normal in $H$, then $N:=\l b,c\r_{p'}M\cong \ZZ_{m}\times\ZZ_2$
is contained in
$\l b,c\r$ and normal in $H$,
and as $x^a=x^{-1}$ for each $x\in\l b,c\r$,
$N$ is normal in $\l H,a\r=X$, the claim is true.

\vskip0.1in
Now, by Theorem~\ref{Praeger}, $\Omega'$ is an arc-transitive
$(\ZZ_{m}\times\ZZ_2)$-cover of
$\Omega'_N$.
Since $p\equiv1~(\mod 3)$,
and $\Omega'_N$ is of valency 3 and order $2p$,
by Lemma~\ref{2p-graphs}, $\Omega'_M\cong\CD(2p,3)$.\qed
\end{proof}

Now, we are ready to complete the proof of Theorem~\ref{Thm-1}.

\vskip0.1in
{\noindent\bf Proof of Theorem~\ref{Thm-1}.}
By assumption, $\Sig$ is $X/K$-arc-transitive of order $2p$,
and so $\Sig$ is listed in Lemma~\ref{2p-graphs}.
To be precise, $\Sig\cong\O_2, \K_{p,p}$ or $\CD(2p,r)$ with $r\mid (p-1)$.

Assume $\Ga\cong\O_2$. All edge-transitive metacyclic covers
of $\O_2$ are determined
in \cite{PH-2},
which are arc-transitive and exactly consist of seven graphs.
Noting that $K$ is not a cyclic group
and a quaternion group,
by \cite[Theorem 1.1]{PH-2},
the tuple $(\Ga,K)$ satisfies part (3) of Theorem~\ref{Thm-1}.


Assume $\Sig\cong\K_{p,p}$.
By Lemma~\ref{K(p,p)}, we have $p<5$,
then by Lemma~\ref{K(3,3)},
we have $K\cong\ZZ_{2n}\times\ZZ_2$ with $n$ odd,
and $\Ga$ is characterized in \cite[Theorem 5.1]{Conder-1},
as in part (1) of Theorem~\ref{Thm-1}.

Assume $\Sig\cong\CD(2p,r)$ with $r\mid (p-1)$.
By Lemma~\ref{CF-(2p,r)},
we have $r=3$ and
part (2) of Theorem~\ref{Thm-1} holds.

Finally, by Lemma~\ref{K-cov},
the graphs $\Ga$ in part (2)(ii) of Theorem~\ref{Thm-1}
are arc-transitive $\ZZ_m\times\ZZ_2$-covers of $\CD(2p,r)$,
and the graph $\CGD(8p,3)$ in part (2)(i) is arc-transitive $\D_4$-cover
of $\CD(2p,3)$ by \cite[Theorem 3.1]{Zhou-Feng},
thus the graphs $\Ga$ in Theorem~\ref{Thm-1}
are really examples.

This completes the proof of Theorem~\ref{Thm-1}.\qed


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}

{ The authors would like to thank the anonymous referee
for the helpful comments.}

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