A new near octagon and the Suzuki tower

We construct and study a new near octagon of order $(2,10)$ which has its full automorphism group isomorphic to the group $\mathrm{G}_2(4){:}2$ and which contains $416$ copies of the Hall-Janko near octagon as full subgeometries. Using this near octagon and its substructures we give geometric constructions of the $\mathrm{G}_2(4)$-graph and the Suzuki graph, both of which are strongly regular graphs contained in the Suzuki tower. As a subgeometry of this octagon we have discovered another new near octagon, whose order is $(2,4)$.


Introduction and overview
A near 2d-gon with d ∈ N is a partial linear space S that satisfies the following properties: (NP1) The collinearity graph of S is connected and has diameter d.
(NP2) For every point x and every line L there exists a unique point π L (x) incident with L that is nearest to x.
A near polygon is a near 2d-gon for some d ∈ N. Near polygons were introduced by Shult and Yanushka in [13] as geometries related to certain line systems in Euclidean spaces. The class of near 4-gons coincides with the class of possibly degenerate generalized quadrangles. Generalized quadrangles belong to the family of generalized polygons, an important class of point-line geometries introduced by Jacques Tits in [16]. It is well known that every generalized 2d-gon is a near 2d-gon. A near polygon is said to have order (s, t) if it has precisely s + 1 points on each line and t + 1 lines through each point. An important class of near polygons with an order is that of the regular near polygons, which are related to the distance regular graphs (see Section 6.4 of [5]). A famous example of a regular near octagon (d = 4) is the near octagon HJ associated with the sporadic simple group J 2 of Hall and Janko (cf. [6]). It can be constructed by taking the 315 central involutions of the group J 2 as points and taking the lines to be the three element subsets {x, y, xy} where the central involutions x and y commute. Therefore, the Hall-Janko near octagon is a commuting involution graph of the sporadic simple Hall-Janko group J 2 .
The five simple groups corresponding to these graphs are L 3 (2) < U 3 (3) < J 2 < G 2 (4) < Suz, which we denote by G i for i = 0, . . . , 4, and the full automorphism group of the graph Γ i is the split extension of G i by the cyclic group of order 2, which is commonly denoted as G i : 2.
In this paper, we associate near polygons S i for i = 0, 1, 2, 3 to the Suzuki tower where each near polygon in the sequence is a subgeometry of the next one. These near polygons have the groups G i : 2 as their full automorphism groups. Moreover, they can be used to construct all the graphs in the Suzuki tower. The near polygons S 0 , S 1 and S 2 are already known and they are respectively isomorphic to the dual of the double of the Fano plane, the dual split Cayley hexagon H D (2) and the Hall-Janko near octagon HJ. We construct a new near octagon G = S 3 of order (2, 10) as a particular kind of commuting involution graph of the group G 2 (4) : 2. On the elements of the line spread S * , a generalized hexagon can be defined as part (1) of the following result shows.
Main Result 2. (1) Let Q * denote the set of all subgeometries of G isomorphic to the generalized quadrangle W (2) (the so-called quads of G). Then the point-line geometry with point set S * , line set Q * and natural incidence relation (i.e. containment) is isomorphic to the dual split Cayley generalized hexagon H D (4).
(2) Let S ⊆ S * and Q ⊆ Q * such that (S, Q) is a full subgeometry of (S * , Q * ) ∼ = H D (4) that is a generalized hexagon of order (4,1). Let P 1 denote the set of points incident with a line of S and L 1 the set of lines incident with a quad of Q. Then G 1 = (P 1 , L 1 ) is a full subgeometry of G = (P, L) that is a near octagon of order (2,4).
The near octagon G 1 of order (2,4) is in fact also a new near octagon. In [9], De Wispelaere and Van Maldeghem showed that the point-line dual of HJ has a full embedding in the split Cayley hexagon H(4). By making use of the fact that (S * , Q * ) ∼ = H D (4), we will give another proof for this fact (see Lemma 4.15 Then this graph is isomorphic to the Suzuki graph. In Section 3, we prove that G is a near octagon and discuss the structure of G with respect to a given point. This structure will be described by means of a so-called suborbit diagram. In Section 4 we prove several geometrical properties of G. The properties related to the line spread S * and the quads of G are discussed in Subsection 4.1 where we also prove Main Result 2. In Subsection 4.2 we classify all suboctagons of G that are isomorphic to the Hall-Janko near octagon HJ, and in Subsection 4.3 we determine the full automorphism group of G, hereby completing the proof of Main Result 1. With the help of the derived geometrical properties we then prove Main Results 3 and 4 in Section 5. Our initial explorations of the properties of the new near octagon happened in a (computer) model for G which was quite different from the one given in Main Result 1.
We first constructed the new near octagon as a subgeometry of the so-called valuation geometry of HJ. Examination of the properties 1 of this new near octagon showed that the full automorphism group was most likely isomorphic to G 2 (4) : 2. Looking for a model of G in the same spirit as the model for HJ (using involutions) was successful and ultimately resulted in the (more symmetric) description given in Main Result 1. In appendix A, we discuss the first model we had for this new near octagon, and show that it is indeed isomorphic to the model presented in Main Result 1.

Preliminaries
A point-line geometry S is a triple (P, L, I), where P is the non-empty point set, L is the line set and I ⊆ P × L is the incidence relation. S is called a partial linear space if any pair of distinct points is incident with at most one line. In this case we may identify each line with the subset of points it is incident with and replace I with set inclusion. By abuse of notation, we would then denote S by (P, L). The distance between two points x 1 and x 2 of a point-line geometry S will always be measured in its collinearity graph. This distance will be denoted by d S (x 1 , x 1 ), or shortly by d(x 1 , x 2 ) if no confusion could arise. If x is a point and i ∈ N, then Γ i (x) denotes the set of points of S at distance i from x. If X 1 and X 2 are two nonempty sets of points, then d(X 1 , X 2 ) denotes the minimal distance between a point of X 1 and a point of X 2 . If X is a nonempty set of points and i ∈ N, then Γ i (X) denotes the set of points at distance i from X.
Let X be a nonempty set of points of a partial linear space S. It is called a subspace if every line meeting X in at least two points is completely contained in X. It is called geodetically closed or convex if every point on every shortest path between two points of X is contained in X. A line spread of S is a set of pairwise disjoint lines that cover all points of S. An ovoid of S is a set of pairwise noncollinear points of S that cover all lines.
A point-line geometry S = (P, L, I) is a subgeometry of another point-line geometry S ′ = (P ′ , L ′ , I ′ ) if P ⊆ P ′ , L ⊆ L ′ and I = I ′ ∩ (P × L). A subgeometry is called full if for every line L in L the set {x ∈ P : x I L} is equal to {x ∈ P ′ : x I ′ L}. If d S (x, y) = d S ′ (x, y) for every two points x, y in P, then we will say that S is isometrically embedded into S ′ .
A quad of a near polygon is a convex subspace Q of diameter 2 such that the full subgeometry determined by those points and lines that are contained in Q is a nondegenerate generalized quadrangle. Sufficient conditions for the existence of quads were given by Shult and Yanushka [13]. In Proposition 2.5 of that paper it was shown that if a and b are two points of a near polygon at distance 2 from each other, and if c and d are two common neighbours of a and b such that at least one of the lines ac, ad, bc, bd contains at least three points, then a and b are contained in a unique quad. This quad coincides with the smallest convex subspace containing a, b and consists of all points of the near polygon which have distance at most 2 from a, b, c and d. A point-quad pair (x, Q) in a near polygon is called classical if there exists a unique point x ′ in Q such that d(x, y) = d(x, x ′ ) + d(x ′ , y) for all y in Q, and ovoidal if the points of Q that are nearest to x form an ovoid of Q. A quad Q is called classical if the pair (x, Q) is classical for every point x. In [13,Proposition 2.6], it was shown that in a near polygon in which each line is incident with at least three points, every point-quad pair is either classical or ovoidal.
A near 2d-gon S with d ≥ 2 is called regular if it has order (s, t) and there exist constants t i , i ∈ {0, . . . , d}, such that for every pair of points x and y at distance i, there are precisely t i + 1 lines through y containing a point at distance i − 1 from x. Clearly, t 0 = −1, t 1 = 0 and t d = t. We will say that S is regular with parameters (s, t; t 2 , . . . , t d−1 ). A generalized 2d-gon is a regular near 2d-gon with parameters (s, t; 0, . . . , 0). The Hall-Janko near octagon is a regular near octagon with parameters (2, 4; 0, 3). In fact it is the unique regular near octagon with those parameters, as proved by Cohen and Tits in [7].
For all the group theoretical notations we refer to the ATLAS (cf. [8]). An involution a of a group G is called central if there exists a Sylow 2-subgroup H of G such that a ∈ C G (H), or equivalently if the centralizer of a contains a Sylow 2-subgroup. It is well known that the group G 2 (4) : 2 has J 2 : 2 as a maximal subgroup of index 416 and the group J 2 : 2 has G 2 (2) as a maximal subgroup of index 100. The conjugacy class of central involutions of the groups G 2 (4) : 2, J 2 : 2, and G 2 (2), are all denoted by the symbol 2A in the ATLAS. If H ∼ = G 2 (2), K ∼ = J 2 : 2 and G ∼ = G 2 (4) : 2 are such that H < K < G and if Σ H , Σ K , Σ G denote the corresponding conjugacy classes of central involutions, then There exists a natural bijective correspondence between the subgroups of G = G 2 (4) : 2 isomorphic to J 2 : 2 and the subgroups isomorphic to J 2 . Every subgroup isomorphic to J 2 : 2 contains a unique J 2 -subgroup, namely its derived subgroup. Conversely, from AT-LAS information we see that every J 2 -subgroup K of G must be contained in a (maximal) subgroup isomorphic to J 2 : 2, and such a maximal is uniquely determined by K as it necessarily coincides with the normalizer of K inside G.

Construction of the near octagon G
The group G = G 2 (4) : 2 has precisely three conjugacy classes of involutions. The class 2A consists of 4095 involutions all of which are central and contained in the derived subgroup G ′ ∼ = G 2 (4). A computer model of the group G can be easily constructed using the computer programming language GAP [10]. All claims of the present section have been verified using such a computer model, see [3].
Let P denote the elements of the class 2A and ω a fixed element in P. The group G acts transitively on P by conjugation. Let G ω be the stabilizer of ω under this action. Then the action of G ω on P partitions P into eight orbits, which we refer to as suborbits of the action of G on P. The suborbits are of sizes 1, 2, 20, 40, 320, 640, 1024, 2048, and we label them There are precisely 62 elements of P \ {ω} that commute with ω and they lie  Figure 1 which explains the choice of notation for these suborbits. Suborbits containing central involutions at distance i from ω in the geometry have been labelled O i * where * is a, b or void.
In the literature suborbit diagrams for finite simple groups where adjacency is defined by commutativity have been studied (see for example [1]). For drawing the suborbit diagram we have used similar conventions as in [1]. In our case adjacency involves both commutativity and a condition on the index of certain normalizers. Each of the eight big nodes of the diagram denotes a suborbit and an edge between two such nodes denotes that there is a line that intersects both suborbits. A smaller node on each edge denotes a line and the two accompanying numbers denote the number of points of the line that lie in the suborbits it intersects. Each number on a big node denotes the number of lines through a given point in that suborbit going to another suborbit.
We would be using the suborbit diagram in most of our arguments where by "suborbit diagram with respect to x" we would mean that ω = x and all the suborbits are defined by the centralizer of the involution x of G 2 (4) : 2. If we wish to explicitly indicate the involution x with respect to which the suborbits are considered, we will use the notations suborbit diagram that every other involution is at distance at most 4 from x. Therefore the point-line geometry is connected with diameter 4. Now let L be any line, then from the suborbit diagram there exists an i ∈ {0, 1, 2, 3} such that L intersects O i in one point and O i+1 in two points. Therefore there exists a unique point on L nearest to x. Since there are exactly 22 neighbours of x, and since the automorphism group acts transitively on points, we get that the point-line geometry is a near octagon of order (2, 10).

Properties of the near octagon G
In this section, we derive several properties of the near octagon G. All of these will be derived from the information provided in Section 3. Proof. By the suborbit diagram, every central involution is contained in a unique element of S * , implying that S * is a line spread of G.

The line spread S * and the quads of G
If x is a point of G, then we denote by L x the unique line through x belonging to S * .
Our next aim will be to determine all quads of G. All these quads are isomorphic to the generalized quadrangle W (2), which is the unique generalized quadrangle of order (2, 2) (cf. [12]). Proof. The points x and y are contained in a quad if and only if x and y have at least two common neighbors. By the suborbit diagram, we know that this happens precisely when y ∈ O 2a (x). If y ∈ O 2a (x), then x and y have precisely three common neighbors, showing that the unique quad through x and y has order (2, 2), necessarily isomorphic to W (2). Moreover, one of the three common neighbors of x and y lies in O 1a (x), showing that L x is contained in the quad.  Proof. Note first that there is at most one quad through M. Indeed, by Corollary 4.3, every quad through M must also contain the line L x , and there is at most one quad through two distinct intersecting lines.
In the suborbit diagram (see Figure 1)  Proof. There are 10 lines through x distinct from L x , and each of these lines is contained in a unique quad (necessarily containing L x ). This gives rise to 10 quads through x, but each of them is counted twice as each of them contains two lines through x distinct from L x . Lemma 4.6. If Q is a quad and L a line of the spread S * containing a point at distance i from Q, then i ∈ {0, 1, 2} and L is completely contained in Γ i (Q).
Proof. We may suppose that i = d(L, Q). Let x ∈ L and y ∈ Q such that d(x, y) = i. By Corollary 4.3, L y is contained in Q.
If i = 0, then L is completely contained in Q by Corollary 4.3. So, we may suppose that i ≥ 1.
Suppose i = 1. Then L x = L is disjoint from Q by Corollary 4.3. By Lemma 4.4, there exists a unique quad through xy. Since this quad contains the lines L x and L y , every point of L x is collinear with a unique point of L y (which belongs to Q).
Suppose i = 2. Let z ∈ Γ 1 (Q) be a common neighbor of x and y. The unique quad through L z ⊆ Γ 1 (Q) and zx contains the line L x , showing that every point of L x is collinear with a point of L z ⊆ Γ 1 (Q), which implies that every point of L x has distance at most and hence precisely 2 from Q.
Suppose i ≥ 3. Since the distance from a point to Q is at most 3, we must have that L ⊆ Γ 3 (Q). Every point of L must then be ovoidal with respect to Q. By (NP2), every point of Q lies at distance 3 from a unique point of L, showing that the three ovoids of Q determined by the points of L form a partition of Q. This is however impossible, as the generalized quadrangle W (2) has no partition in ovoids.

Lemma 4.7. Every quad of G is classical.
Proof. Let Q be a quad of G. By Lemma 4.6, every point has distance at most 2 from Q. Every point at distance at most 1 from a quad of a general near polygon is classical with respect to that quad. In particular, every point of Let x be a point in Γ 1 (Q) and x ′ the unique neighbour of x in Q. If L is a line through x contained in Γ 1 (Q) then the points of Q collinear with a point of L form a line L ′ . The lines L and L ′ are contained in a quad, which necessarily coincides with the unique quad So, through x there is a unique line meeting Q, two lines contained in Γ 1 (Q) and eight lines meeting Γ 2 (Q) (necessarily in two points). Since |Γ 1 (Q)| · 8 · 2 = 3840 = |Γ 2 (Q)|, we must have that every point of Γ 2 (Q) is collinear with a unique point of Γ 1 (Q) and hence at distance 2 from a unique point in Q. The quad Q must therefore be classical.
The relation defined on pairs (x, y) ∈ P × P by the condition if and only if d(x, y) = 2 and x, y commute (regarded as involutions). (The fact that the relation is symmetric also follows from Lemma 4.2.) The following lemma shows that also the relation defined on pairs (x, y) ∈ P × P by the condition y ∈ O 3a (x) is symmetric.
(3) there is a quad through x meeting a line through y; (4) there is a quad through y meeting a line through x.
Proof. By the suborbit diagram and Lemma 4.2, (1) and (3) are equivalent, as well as (2) and (4). So, by symmetry it suffices to show that (3) implies (4). Suppose Q is a quad through x and L is a line through y meeting Q in a point z. The line L z is contained in Q and contains a point u collinear with x. The unique quad through L = zy contains the line L z and meets the line xu through x. (2) L meets the suborbits O 1b and O 2a ; (3) L meets the suborbits O 2b and O 3a ; (4) L meets the suborbits O 3b and O 4 .
Suppose L contains a point x ∈ O 1b . Then the unique quad through the line ωx contains L and is completely contained On the other hand, suppose that L contains a point y ∈ O 2a . Then ω and y are contained in a unique quad which contains the line L.
In the sequel, we will therefore assume that L is disjoint Suppose As before, we denote by Q * the set of quads of G. Proof. The collinearity graph of H D (4) is isomorphic to the graph whose vertices are the 1365 long root subgroups of G 2 (4) = G ′ , with two distinct long root subgroups being adjacent whenever they commute. The lines of H D (4) then correspond to the maximal cliques (of size 5) of this collinearity graph. Since every line of (S * , Q * ) contains five points, it thus suffices to prove that the collinearity graphs of (S * , Q * ) and H D (4)   Proof. Put δ ′ := d(K, M) and let x 0 , x 1 , . . . , x δ ′ be a path of length δ ′ connecting a point If two consecutive lines L i and L i+1 are distinct, then they are contained in the unique quad through x i x i+1 and so every point of L i is collinear with a unique point of L i+1 . But since δ ′ is the smallest distance between a point of K and a point of M, two consecutive lines are L i and L i+1 must be distinct (otherwise we could construct a shorter path), and every point of K has distance δ ′ from a necessarily unique point of M. So, in order to prove the lemma, it suffices to show that δ = δ ′ . Since there exists a path of length δ ′ in (S * , Q * ) connecting K and M, we have δ ≤ δ ′ . So, it suffices to show that δ ≥ δ ′ . Suppose U 0 , U 1 , . . . , U δ is a path of length δ in (S * , Q * ) connecting the lines U 0 = K and U δ = M. For every two consecutive lines U i and U i+1 , we know that every point of U i is collinear with a unique point of U i+1 , implying that there exists a point of M at distance at most δ from a point of K. This implies that δ ≥ δ ′ , as we needed to show.
The following is an immediate corollary of Lemma 4.11.  Proof. Let L ∈ L 1 . Then L ⊆ Q for a certain quad Q ⊆ Q. The lines of S * contained in Q all belong to S, implying that every point of L belongs to P 1 . So, G 1 = (P 1 , L 1 ) is a full subgeometry of G = (P, L).
If x and y are two points of P 1 which are collinear in G, then the lines L x ∈ S and L y ∈ S are equal or collinear in (S * , Q * ) and hence also in (S, Q). There must exist a quad Q ∈ Q containing L x and L y . The lines of S contained in Q cover Q, showing that every point of the line xy is contained in P 1 . So, P 1 is a subspace of G and the full subgeometry induced on P 1 is precisely G 1 .
Note that any subhexagon of order (4, 1) of H D (4) is isometrically embedded into H D (4). So, if K and L are two lines of S, then the distance between K and L is the same in the geometries (S, Q) and (S * , Q * ). Corollary 4.12 then implies that the distance between two points x ∈ P 1 and y ∈ P 1 is the same in the geometries G and G 1 . So, Property (NP2) in the definition of near polygon remains valid for G 1 . By taking suitable points on opposite lines belonging to S, we see that the diameter of G 1 is also 4. So, G 1 is a near octagon.
Every point x ∈ P 1 is contained in two quads of S which intersect in the line L x . So, there are precisely five lines of L 1 through x, showing that the near octagon G 1 has order (2, 4).

The Hall-Janko suboctagons of G
In this subsection, we classify all Hall-Janko suboctagons of G. These are (full) subgeometries of G that are isomorphic to HJ. We will show that there are 416 such subgeometries and that all of them are isometrically embedded into G. In the following lemma, we already construct all these 416 subgeometries from the 416 (maximal) subgroups of G 2 (4) : 2 isomorphic to J 2 : 2. Before proceeding to prove that every Hall-Janko suboctagon is as described in Lemma 4.14, we first give an alternative proof of a result of [9], stating that the point-line dual HJ D of HJ has a full embedding into the split Cayley hexagon H(4). Proof. Let H be a maximal subgroup of G isomorphic to J 2 : 2. Then S H is a full subgeometry isomorphic to HJ. Every line L of S H is contained in a unique quad Q L (as L ∈ S * ). As any two involutions of Q L ∩ H commute, Q L ∩ H is at most a line of H, implying that L = Q L ∩ H. So, if L 1 , L 2 , . . . , L 5 are the five lines of S H through a given point x, then the quads Q L 1 , Q L 2 , . . . , Q L 5 are mutually distinct and hence are all the five quads through L x . This implies that the maps x → L x , L → Q L define a full embedding of the dual of S H into the dual of (S * , Q * ), which is isomorphic to H(4).
In the sequel, H will denote an arbitrary Hall-Janko suboctagon. We will derive several properties of H that will enable us to prove that there are at most (and hence precisely) 416 Hall-Janko suboctagons. Proof. Obviously, this is true if d H (x, y) ≤ 1. So, suppose that d H (x, y) = 2. Then d G (x, y) ≤ 2. If d G (x, y) = 1 then we would get a triangle which contradicts (NP2). Therefore, d G (x, y) = 2.      Proof. We also have d G (x, y) = 2. Let x ′ ∈ H be a common neighbour of x and y. If y ∈ O 2a (x), then the unique quad through x and y would contain the intersecting lines xx ′ and x ′ y, which would be in violation with Lemma 4.19. Therefore, y ∈ O 2b (x). Proof. Let x and y be two opposite points of G and suppose the Hall-Janko octagon H contains x and y. We will show that H is uniquely determined by x and y. In this proof all suborbits are considered with respect to the point x. By Lemma 4.22, the distance between two points of H is the same in the geometries H and G.
There are five lines through y inside H that contain a point at distance 3 from x. By Lemma 4.17 all of these lines must intersect O 3b . By the suborbit diagram and Lemma 4.9, there are exactly six such lines through y and one of them is in S * . By Lemma 4.20, the line belonging to S * cannot be contained in H. Therefore the five lines of H through y, going back to x are uniquely determined by x and y. Now let y ′ ∈ O 3b be a point on one of these five lines and Q the unique quad through yy ′ and L y ′ = yy ′ . By Lemma 4.9, L y ′ meets O 4 . By Lemma 4.21 the third line of Q through y ′ , call it M y ′ , doesn't lie in H. We claim that M y ′ intersects O 2b . Indeed, as the point x is classical with respect to Q, the unique point u in Q nearest to x lies at distance 2 from x and is collinear with y ′ . Therefore, u ∈ O 2b and M y ′ = y ′ u. The four lines of H through y ′ that go back to x are now uniquely determined. Indeed, by Lemma 4.23, each of the four lines of H through y ′ meets O 2b . But by the suborbit diagram, there are precisely five such lines. Moreover, one of these five lines is the line M y ′ and we already know that it cannot be a line of H. Now, let y ′′ ∈ O 2b be a point on one of these four lines. By the suborbit diagram there is a unique line through y ′′ containing a point y ′′′ in O 1b , which must necessarily be in H. Moreover, there is a unique line through y ′′′ that contains x.
So far, we have proved that given any point y in H with d H (x, y) = 4, all shortest paths between x and y in H are uniquely determined by x and y. Moreover, all points at distance 4 from x that are collinear with y are uniquely determined. These properties in fact imply that the whole of H is uniquely determined. Indeed, the subgraph of the collinearity graph induced on the set Γ 4 (x) ∩ H is connected (see Step 1 of the proof of Theorem 3 in [7]), and every shortest path between x and a point of H can be extended to a shortest path between x and a point of Γ 4 (x) ∩ H.  Proof. Let H be a Hall-Janko suboctagon of G and x a point not contained in H. Say x has two neighbours y, z in H. Then by Lemma 4.22 d H (y, z) = 2 and hence there is a common neighbour of y, z inside H. This means that there is a quad through y, z whose intersection with H contains a pair of intersecting lines, contradicting Lemma 4.19. Therefore, if x has a neighbour in H then it is unique. Now we can show that x has a neighbour in H by a simple counting. There are six lines out of the eleven through each point in H that are not contained in H, giving us a total of 12 · 315 points of G at distance 1 from H, as they all must be distinct. Adding this to the number of points in H we get 315 · 12 + 315 = 4095 which is the total number of points in G.  Proof. We consider the following two cases: 1. The point x is collinear with y. Let x ′ = π H (x) = π H (y) and z ∈ Γ 4 (x ′ ) ∩ H. Since {x, y, x ′ } is a line, by (NP2), either d(z, x) = 3 and d(z, y) = 4, or d(z, x) = 4 and d(z, y) = 3. In either case z belongs to only one of H ∩ Γ 4 (x), H ∩ Γ 4 (y).

The point
x is not collinear with y. Consider the suborbit diagram with ω equal to the common projection of x and y.

The automorphism group of G
In this subsection, we show that the full automorphism group of G is isomorphic to G = G 2 (4) : 2. Proof. Each g ∈ G determines an automorphism of G: if x is a central involution and g ∈ G, then x g = g −1 xg is again a central involution. Since the central involutions generate the group G ′ = G 2 (4) and C G (G ′ ) = 1, the action of each g ∈ G \ {e} is faithful. Proof. The automorphism θ permutes the quads of G and hence the lines of G that can be obtained as intersections of two quads. Proof. Let x be an arbitrary point of G, L = {x, y, z} a line through x not belonging to S * and Q the unique quad through L. The lines of S * contained in Q determine a spread of the W (2)-quad Q. Inside Q, it is easily seen that L is the unique line of Q meeting L x , L y and L z . From L θ x = L x , L θ y = L y and L θ z = L z , it then follows that x θ = x. Proposition 4.31. The full automorphism group of G is isomorphic to G 2 (4) : 2.
It is possible to give another proof of Proposition 4.31 based on the following lemma. Proof. The action of θ on the point set of S H is given by conjugation by a suitable element of H ∼ = J 2 : 2. This conjugation also determines an automorphism of G. To show that θ extends to at most one automorphism of G, we must show that every automorphism ϕ of G that fixes each point of S H must be trivial. But this is implied by Lemma 4.27.

The Suzuki tower
Let S 0 , S 1 , S 2 , S 3 be the near polygons and Γ 0 , Γ 1 , Γ 2 , Γ 3 , Γ 4 the graphs of the Suzuki tower as mentioned in Section 1. Then we know that S 0 = H(2, 1), S 1 = H D (2), S 2 = HJ and S 3 = G, where H(2, 1) is the unique generalized hexagon of order (2, 1). We define S −1 to be the partial linear space on nine points and four lines obtained from the (3 × 3)grid by removing two disjoint lines (and keeping the points incident with these two lines). We define S ′ −2 to be a line with three points and S ′′ −2 to be a coclique of size 3 (no lines). The graphs Γ i with i = 0, 1, 2, 3 can all be obtained in a similar way from the near polygons S i and some of their subgeometries. It can easily be verified that Γ 0 is isomorphic to the graph whose vertices are the subgeometries of S 0 isomorphic to S −1 , where two such subgeometries are adjacent whenever they intersect in a subgeometry isomorphic to S ′ −2 or S ′′ −2 . The graph Γ i with i = 1, 2 is known to be isomorphic to the graph whose vertices are the subgeometries of S i isomorphic to S i−1 , where two subgeometries are adjacent whenever they intersect in a subgeometry isomorphic to S i−2 . We prove an analogous property 2 for the graph Γ 3 .
Lemma 5.1. The G 2 (4)-graph Γ 3 is isomorphic to the graph Γ ′ whose vertices are the Hall-Janko suboctagons of G, where two Hall-Janko suboctagons are adjacent whenever they intersect in a subgeometry isomorphic to H D (2).
Proof. The G 2 (4)-graph is the graph whose vertices are the maximal subgroups of G 2 (4) isomorphic to J 2 , where two such maximal subgroups are isomorphic if they intersect in a subgroup isomorphic to G 2 (2) ′ ∼ = U 3 (3).
It is well-known that the Hall-Janko near octagon HJ has 100 subhexagons isomorphic to H D (2), and that these are in bijective correspondence with the 100 maximal subgroups of J 2 isomorphic to G 2 (2) ′ . The points of a subhexagon are the central involutions contained in the corresponding maximal subgroup. Moreover, these central involutions generate the maximal subgroup.
For every subgroup H of G 2 (4), denote by Σ H the set of central involutions contained in H. If H ∼ = J 2 , then the geometry S H induced on the subspace Σ H is isomorphic to HJ. By Lemma 4.25, the map H → S H defines a bijection between the 416 maximal subgroups of G 2 (4) isomorphic to J 2 and the 416 Hall-Janko suboctagons of G. We show that this map defines an isomorphism between the G 2 (4)-graph and the graph Γ ′ . Take two mutually distinct subgroups H 1 and H 2 of G 2 (4) isomorphic to J 2 .
If H 1 and H 2 are two adjacent vertices of the G 2 (4)-graph, then the subgeometries S H 1 and S H 2 intersect in a subgeometry whose point set is Conversely, suppose that S H 1 and S H 2 intersect in a subgeometry isomorphic to H D (2). Then Σ H 1 ∩ Σ H 2 contains all central involutions that are contained in a certain G 2 (2) ′subgroup K i of H i ∼ = J 2 , i = 1, 2. Since all these central involutions generate K i , the groups K 1 and K 2 are equal, say to K. As K is a maximal subgroup of both H 1 and H 2 , we have K = H 1 ∩ H 2 , i.e. H 1 and H 2 are adjacent in the G 2 (4)-graph.
Lemma 5.1 is precisely Main Result 3. The graphs Γ 0 , Γ 1 , Γ 2 , Γ 3 of the Suzuki tower can all be constructed from the near polygons S 0 , S 1 , S 2 , S 3 and some of their subgeometries. Main Result 4 which we will now prove says that this is also true for the remaining graph Γ 4 in the Suzuki tower. In fact, the construction given in Main Result 4 is a translation (in terms of subgeometries of S 3 = G) of the original construction of the Suzuki graph [15]. We wish to note that it is also possible to give similar constructions for the other graphs of the Suzuki tower by translating their original constructions in terms of substructures of a suitable S i . We will omit these other constructions here.
Let ∞ be an extra symbol. Let S be the conjugacy class of 2-subgroups of type 2A if i = 1, 2 or 3. In the remaining case, i = 4, let S be the conjugacy class of are adjacent as vertices of ∆, a vertex x ∈ S is adjacent to a vertex v ∈ V (∆) if a non trivial element of the subgroup corresponding to x fixes v, and two vertices x, y in S are adjacent if x, y considered as subgroups of H do not commute but there exists a z ∈ S that commutes with both of them.
In the case i = 4, ∆ is the G 2 (4)-graph and H ∼ = G 2 (4) : 2. We know that the vertices of ∆ can be put in 1-1 correspondence with the Hall-Janko suboctagons of G and that the elements of S (the long root subgroups) can be put in 1-1 correspondence with the lines of the spread S * . Two long root subgroups are adjacent whenever they do not commute, but there exists a long root subgroup that commutes with both. In terms of properties of G, this means that the corresponding lines of S * must lie at distance 2 from each other in the near polygon. By Lemma 4.32 we know that every automorphism stabilizing a Hall-Janko suboctagon H must be a conjugation by an element of the J 2 : 2-subgroup corresponding to H. From this it follows that if x ∈ S and v ∈ V (∆), then a non-trivial element of the subgroup corresponding to x fixes v if and only if the spread line corresponding to x intersects the Hall-Janko suboctagon corresponding to v. This all implies that the graph as defined in Main Result 4 should be isomorphic to the Suzuki graph Γ 4 .
Remark: There are other known strongly regular graphs that can be constructed from substructures of these near polygons in a similar way as Main Results 3 and 4. For example, it is known that the Hall-Janko near octagon contains 280 copies of the generalized octagon of order (2, 1), denoted as GO(2, 1), as convex subgeometries (cf. Proposition 4.7 in [17]), with every pair of distinct GO(2, 1)'s intersecting in 5 or 15 points. Computations showed that the graph defined on these 280 suboctagons, where adjacency is defined by intersection in 15 points is an srg(280, 36, 8, 4) (this fact was communicated to Andries Brouwer who included it on the website [4]).
Computations with subgeometries of G also showed that if N 1 is a Hall-Janko suboctagon and N 2 a G 1 -suboctagon then N 1 ∩ N 2 is either isomorphic to H(2, 1) or GO(2, 1). These intersections give us 56 suboctagons of G 1 isomorphic to GO(2, 1), with every pair of distinct GO(2, 1)'s intersecting in 5 or 9 points. We can define a graph on these 56 suboctagons by defining adjacency as intersection in 9 points. Computations revealed that this is an srg(56, 10, 0, 2) necessarily isomorphic to the unique strongly regular graph with those parameters, the well known Sims-Gewirtz graph. We can also construct the graph srg(162, 56, 10, 24), which is the second subconstituent of the McLaughlin graph, by taking the elements of {∞}, A and B as vertices where A is the set of 56 sub GO(2, 1)'s of G 1 and B the set of 105 lines of S * that are contained in G 1 . Then join ∞ to all vertices in A, join two distinct vertices of A if the corresponding suboctagons intersect in 9 points, joint a vertex of A to all vertices of B that correspond to a line intersecting the suboctagon, and join two vertices of B if the lines are at distance 2.

A The original construction of the near octagon G
In this appendix, we give another description of the near octagon G. This description was the first description we had for this new near octagon and arose while the authors were studying near polygons that contain HJ as an isometrically embedded subgeometry. We define a valuation of HJ as a map from the point set of HJ to N satisfying:  For the purpose of studying near polygons containing HJ as an isometrically embedded subgeometry, the authors determined all valuations of HJ with the aid of GAP, see [3]. It    can be found as the (i + 1)-th entry in "value distribution". Subsequently, we have determined the possible line types for the lines of V, together with information saying how many lines of each type are incident with a given point of Type T ∈ {A, B, C, D, E}. This information can be found in Table 2. Now, take the subgeometry V ′ of order (2, 10) of V whose points are the valuations of Type A, B, C, and whose lines are the lines of Type AAA, ABB, ACC, BBC, CCC. Computer computations showed that this is a near octagon (containing HJ as a full suboctagon). Computer computations also revealed that G 2 (4) : 2 was the most likely candidate for the full automorphism group (see Section 1). An attempt to construct the near octagon directly from the group G 2 (4) : 2 was successful and resulted in the description given in Main Result 1. We end this appendix by showing that the geometries G and V ′ are indeed isomorphic.
Proposition A.2. The near octagon G is isomorphic to V ′ .
Proof. Regard HJ as a full subgeometry of G. Then HJ is isometrically embedded into G by Lemma 4.22. By Lemma A.1, every point x of G will induce a valuation f x of HJ.
This valuation is of Type A if and only if x belongs to HJ. By Lemma 4.26, each induced valuation has a unique point with value 0. So, all induced valuations have Type A, B or C. By Lemma 4.27, all induced valuations are distinct, implying that the 4095 induced valuations are precisely the 4095 valuations of HJ that have Type A, B or C. Now, every point of G is incident with precisely 11 lines. By looking at the columns "A" and "C" of Table 2, we see that all lines of V of Type AAA, ABB, ACC, BBC and CCC should be induced (in the sense of Lemma A.1(2)). The number of such lines of V is equal to 315·5 3 + 315 · 1 + 315 · 5 + 3150 · 1 + 3150·9 3 = 15015. Since G has 4095·11 3 = 15015 lines, we see that the lines of V that are induced are precisely the lines of Type AAA, ABB, ACC, BBC and CCC. We can now conclude that G and V ′ are isomorphic.