On groups associated with the affine subgroups of $Sp_{2n}(2)$

Abstract

The symplectic group $Sp_{2n}(2)$ has an affine maximal subgroup of structure $ASp_n=2^{2n-1}{:}Sp_{2n-2}(2)$ which is a split extension of an elementary abelian 2-group $N=2^{2n-1}$ by $G=Sp_{2n-2}(2)$. The vector space $N=2^{2n-1}$ and its dual $N^{*}$ are not equivalent as $2n-1$ dimensional G-modules over GF(2). Therefore, a split extension of the form $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)\ncong N{:}Sp_{2n-2}(2)$ exists. In this paper, it will be shown that $\overline{G}_n\cong \text {Aut}(2^{2n-2}{:}Sp_{2n-2}(2))= \left( 2^{2n-2}{:}Sp_{2n-2}(2)\right) {:} 2$ for $n\ge 3$. Moreover, the ordinary irreducible characters of $\overline{G}_n$ are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set Irr$(\overline{G}_5)$ of the group $\overline{G}_5=2^9{:}Sp_{8}(2)$ which is associated with the affine subgroup $ASp_5=2^9{:}Sp_{8}(2)$ of $Sp_{10}(2)$.

1 Introduction

Symplectic groups are constructed by defining some bilinear form on the underlying vector space and then taking all the form-preserving automorphisms of the space. The subgroups of symplectic groups $Sp_{2n}(2)$ which fix a non-zero vector of the underlying symplectic space are called affine subgroups having the structure $ASp_n=2^{2n-1}{:}Sp_{2n-2}(2)$ which is a split extension of $G=Sp_{2n-2}(2)$ by an elementary abelian 2-group $N=2^{2n-1}$. The group $ASp_n$ has 4 orbits on N of lengths 1, 1, $2^{2n-2}-1$ and $2^{2n-2}-1$ with corresponding orbit stabilizers $G, G, 2^{2n-3}{:}Sp_{2n-4}(2)$ and $2^{2n-3}{:}Sp_{2n-4}(2)$. Moreover, $ASp_n$ has 4 orbits of lengths $1, 2^{n-2}(2^{n-1}-1),2^{n-2}(2^{n-1}+1)$ and $2^{2n-2}-1$ on Irr(N) with corresponding point stabilizers $Sp_{2n-2}(2)$, $O^{-}_{2n-2}(2){:}2$, $O^{+}_{2n-2}(2){:}2$ and $2^{2n-3}{:}Sp_{2n-4}(2)$. All of the information above is found in [1] and [11] where the ordinary character tables of the affine subgroups of $Sp_8(2)$ and $Sp_6(2)$ were computed using the Fischer-Clifford matrices technique [6]. The dual space $N^{*}$ of the vector space $N=2^{2n-1}$ is identified with the set Irr(N). Since G has different orbit lengths on N and Irr(N), respectively, it follows that N and $N^{*}$ are not equivalent as $2n-1$ dimensional G-modules over GF(2).

From the discussion above, we deduce that a split extension of the form $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)$ exists and it is not isomorphic to $ASp_n$. Obviously, the actions of $ASp_n$ on N and Irr(N) are reversed for the actions of $\overline{G}_{n}$ on $N^{*}$ and $Irr(N^{*})$. Hence, $\overline{G}_n$ has 4 orbits $\Gamma _s$ of lengths $|\Gamma _{1}|=1$, $|\Gamma _{2}|=2^{n-2}(2^{n-1}-1)$, $|\Gamma _{3}|=2^{n-2}(2^{n-1}+1)$ and $|\Gamma _{4}|=2^{2n-2}-1$ on the classes of $N^{*}$ with respective point stabilizers $Sp_{2n-2}(2)$, $O^{-}_{2n-2}(2){:}2$, $O^{+}_{2n-2}(2){:}2$ and $2^{2n-3}{:}Sp_{2n-4}(2)$, whereas $\overline{G}_n$ has 4 orbits $\triangle _{s}$ on Irr$(N^*)$ of lengths 1, 1, $2^{2n-2}-1$ and $2^{2n-2}-1$ with corresponding orbit stabilizers as G, G, $2^{2n-3}{:}Sp_{2n-4}(2)$ and $2^{2n-3}{:}Sp_{2n-4}(2)$.

Theorem 1.1

The center $Z(ASp_n)$ of $ASp_n$ is isomorphic to $\mathbb {Z}_2$.

Proof

N has $k=2^{2n-1}$ orbits on the identity coset N of N in $ASp_n$. Furthermore, the k orbits fuse into 4 orbits of lengths $f_j$ under the action of the centralizer $C_{G}(1A)=G$, where $f_1=f_2=1$, $f_3=f_4=2^{2n-2}-1$. Using the coset analysis technique [10, 11], the centralizer orders of the 4 classes $[x_j]$ of $\overline{G}_{n}$ coming from N are given by $|C_{ASp_n}(x_j)|=\frac{k|C_{G}(1A)|}{f_j}= \frac{2^{2n-1}|G|}{f_j}$. If $f_j=1$ then $|C_{ASp_n}(x_j)|=|ASp_n|$ and so we obtain two classes $[x_j]$ of $ASp_n$ containing each one element. Since $Z(Sp_{2n-2(2)})$ is trivial, only the identity class 1A of $Sp_{2n-2}(2)$ contains one element. Hence, by the coset analysis method, no other class $[x_j]$ of $ASp_n$ coming from a non-identity coset Ng, $g \in G$, of N in $ASp_n$ contains one element. Hence, $ASp_n$ has only two classes $[x_j]$ containing one element each. Therefore, the center $Z(ASp_n)$ of $ASp_n$ is isomorphic to $\mathbb {Z}_2$. $\square$

Since $Z(ASp_{n})\triangleleft ASp_{n}$ and $Z(ASp_{n})\triangleleft N$, Corollary 1.2 below follows from Theorem 1.1.

Corollary 1.2

$\frac{ASp_{n}}{Z(ASp_{n})}\cong 2^{2n-2}{:}Sp_{2n-2}(2)\cong \text {Inn}(ASp_{n})$ and therefore $ASp_{n}=2^{^.}(2^{2n-2}{:}Sp_{2n-2}(2))$.

For $n \ge 3$, only one of the four orbits of $\overline{G}_n$ on the classes of $N^{*}$ contains 1 element and also $Z(Sp_{2n-2})$ is trivial. Thus Theorem 1.3 is obtained by using the coset analysis method and a similar argument as in Theorem 1.1.

Theorem 1.3

The center $Z(\overline{G}_n)$ of $\overline{G}_n$ is trivial for $n \ge 3$.

Corollary 1.4

Since $Z(\overline{G})$ is trivial, $\overline{G}_n\cong \text {Inn}(\overline{G}_n)$ for $n \ge 3$.

Remark 1.5

The actions of $ASp_2$ on N and Irr(N) are equivalent and hence $\overline{G}_2\cong ASp_2$. Therefore, $Z(\overline{G}_2)\cong \mathbb {Z}_2$. Moreover, $\overline{G}_2=(2^2{:}Sp(2,2))\times 2\cong S_4\times 2$.

Consider the group $\overline{F}_n=2^{2n-2}{:}Sp_{2n-2}(2)$, which acts transitively on the non-zero vectors of the vector space $N_1=2^{2n-2}$ and hence has two orbits of lengths 1 and $2^{2n-2}-1$. By Brauer’s theorem $\overline{F}_n$ also has two orbits of lengths 1 and $2^{2n-2}-1$ on Irr$(N_1)$ with $Sp_{2n-2}(2)$ and $2^{2n-3}{:}Sp_{2n-4}(2)$ as inertia factors, respectively. Now under the natural homomorphism $\eta {:}N^{*}\rightarrow N_1$ and using the fact that $N^{*}=2\times N_1$ it follows that the pre-image $\eta ^{-1}\left\{ [n_1]\right\}$ of a class $[n_1]$ of $N_1$ is the set of two classes $\left\{ [n_{1_1}],[n_{1_2}]\right\}$ of $N^{*}$, where $[n_{1_1}]$ is taken as $[n_1]$ since $N_1\le N^{*}$. Besides the orbits of length 1 and $2^{2n-2}-1$ which contain the elements of $N_1$, the other $2^{2n-2}$ elements of $N^{*}$ coming from the classes $[n_{1_2}]$ are in two orbits of lengths $2^{n-2}(2^{n-1}-1)$ and $2^{n-2}(2^{n-1}+1)$ under the action of $\overline{G}_n$. Also, the set Irr$(N_1)$ is contained in two orbits of lengths 1 and $2^{2n-2}-1$ because of the action of $\overline{G}_n$ on Irr$(N^{*})$ and since $\overline{F}_n$ is a split extension with an abelian kernel $N_1$ it implies that $|\text {Irr}(\overline{F}_n)|= |\text {Irr}(Sp_{2n-2}(2))| +|\text {Irr}(2^{2n-3}{:}Sp_{2n-4}(2))|$. Moreover, the other two orbits of $\overline{G}_n$ on Irr$(N^{*})$ also have the same lengths of 1 and $2^{2n-2}-1$ with corresponding inertia factors $Sp_{2n-2}(2)$ and $2^{2n-3}{:}Sp_{2n-4}(2)$ and hence $|\text {Irr}(\overline{G}_n)|= 2\times |\text {Irr}(Sp_{2n-2}(2))| +2\times |\text {Irr}(2^{2n-3}{:}Sp_{2n-4}(2))|=2\times |\text {Irr}(\overline{F}_n)|$. It follows that we can write the structure of $\overline{G}_n$ in terms of $\overline{F}_n$ in four possible ways, that is, $\overline{G}_n=\overline{F}_n\times 2$, $\overline{G}_n=2^{\cdot }\overline{F}_n$, $\overline{G}_n=\overline{F}_n{^\cdot }2$ or $\overline{F}_n{:}2$. Since $Z(\overline{G}_n)$ is trivial by Theorem 1.3, it follows that, for $n\ge 3$, $\overline{G}_n$ does not have a normal subgroup of order 2 and hence $\overline{G}_n$ is either isomorphic to $\overline{F}_n{^\cdot }2$ or $\overline{F}_n{:}2$. It is well-known that $Z(\overline{F}_n)$ is trivial and hence Inn$(\overline{F}_n)\cong \overline{F}_n$. Furthermore, it is also known that Out$(Sp_{2n-2}(2))$ is trivial and $N_1=2^{2n-2}$ is an absolutely irreducible module for $Sp_{2n-2}(2)$. With the discussion above, Theorem 1.6 below follows now from the fact that $|\text {Out}(\overline{F}_n)|=|H^1(Sp_{2n-2}(2),N_1)|=2$ (see Table 4.5 in [4]).

Theorem 1.6

$\text {Aut}(\overline{F}_n)=\text {Inn}(\overline{F}_n){.}\text {Out}(\overline{F}_n)\cong \overline{F}_n{:}2=\overline{G}_n$ for $n \ge 3$.

In Sect. 2, the Fischer-Clifford matrices and ordinary irreducible characters of the groups of general structure $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)$ are discussed. As an example, we choose $\overline{G}_5=2^{9}{:}Sp_{8}(2)$ to compute its ordinary character table using the Fischer-Clifford matrices technique. For us to use this method, the conjugacy classes of $2^{9}{:}Sp_{8}(2)$ are computed from the classes of the group $Sp_8(2)$ by the coset analysis method. Having computed the classes of $\overline{G}_5$, the fusion maps of the so-called inertia factors into $Sp_8(2)$ and the Fischer-Clifford matrices of $\overline{G}_5$, the information from these computations is tabulated in the relevant sections of this paper. The ordinary character tables of the inertia factors are found in the GAP library. Constructing the ordinary character table of $\overline{G}_5$ from the aforementioned information via the Fischer-Clifford matrices technique, a $390\times 390$ complex invertible matrix is obtained and is accessible through a link which is provided at the end of this paper. Computations are carried out with the assistance of GAP [7] and MAGMA [2].

2 Fischer-Clifford matrices technique applied to $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)$

In this section, the construction of the Fischer-Clifford matrices of the groups of structure $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)$ is discussed. The matrix M(1A) is generalized for $\overline{G}_n$. The structure of the ordinary character table of $\overline{G}_n$ is given in the format obtained by the Fischer-Clifford matrices method. A more general and detailed treatment of Fischer-Clifford matrices of an extension group is found in [1, 6] or [11]. For recent publications in Fischer-Clifford theory readers are referred to [12] and [13]. The literature [8] and [9] can be consulted for a background in character theory and some of the results used in this section.

2.1 The construction of Fischer-Clifford matrices of $\overline{G}_n$

It follows from Sect. 1 that $\overline{G}_n$ has 4 orbits on Irr$(N^*)$ with corresponding inertia groups $\overline{H}_s=N^*{:}H_s=\left\{ x\in \overline{G}_n | \theta _s^{x} = \theta _s\right\}$, $s=1,2,3,4$, where $\theta _s\in \text {Irr}(N^*)$ are orbit representatives. The inertia factors $H_s$ are $H_1=H_2=Sp_{2n-2}(2)$ and $H_3=H_4=2^{2n-3}{:}Sp_{2n-4}(2)$. Since $\overline{G}_n$ is a split extension and $N^*$ is an elementary abelian 2-group it follows that each $\theta _s$ extends to a $\psi _s\in \text {Irr}(\overline{H_s})$, i.e. $\psi _s\downarrow _{N^*}$ = $\theta _s$, by Mackey’s Theorem (see Theorem 5.1.15 in [11]). Furthermore, by Theorem 5.1.7, Remark 5.1.8 and Theorem 5.1.19 in [11], an ordinary irreducible character $\chi =(\psi _s\overline{\beta })^{\overline{G}_n}$ of $\overline{G}_n$ is obtained by induction of $\psi _s\overline{\beta }\in \text {Irr}(\overline{H_s})$ to $\overline{G}_n$, where $\overline{\beta } \in \text {Irr}(\overline{H}_s)$ such that $N^*\le ker(\overline{\beta })$. Note that $\overline{\beta }\in \text {Irr}(\overline{H_s})$ is a lifting of $\beta \in \text {Irr}(H_s)$ to $\overline{H_s}$. Therefore,

$$\begin{aligned} \begin{aligned} \text {Irr}(\overline{G}_n)&= \bigcup _{s = 1}^{4} \{ ( \psi _s \overline{\beta })^{\overline{G}_n} | \overline{\beta } \in \text {Irr}(\overline{H_s}), N^*\subseteq \text {ker}(\overline{\beta }\} = \bigcup _{s = 1}^{4} \{ ( \psi _s \overline{\beta })^{\overline{G}_n} | \beta \in \text {Irr}(H_s)\}. \end{aligned} \end{aligned}$$

Hence the set Irr$(\overline{G}_n)$ is partitioned into 4 blocks $B_s$ with each block $B_s$ corresponding to an inertia group $\overline{H_s}$. Observe that $|\text {Irr}(\overline{G}_n)|=|\text {Irr}(H_1)|+|\text {Irr}(H_2)|+|\text {Irr}(H_3)|+|\text {Irr}(H_4)|$.

We take $\overline{H_1}$= $\overline{G}_n$ and $H_1$ = $Sp_{2n-2}(2)$. We define the set

$$\begin{aligned} R (g) = \{ (s, y_k) \, | \, 1 \le s \le 4, H_s \cap [g] \not = \emptyset , 1 \le k \le r \},\end{aligned}$$

where $y_k$, $k=1, 2,...,r$, are representatives of conjugacy classes $[y_k]$ of $H_s$ that fuse into a class [g] of $H_1$. Let $y_{l_k}$ be representatives of the conjugacy classes of $\overline{H}_s$, where each $y_{l_k}$ has $y_k$ as an image under the homomorphism $\overline{H_s}\longrightarrow H_i$ whose kernel is $N^*$. Also, we let $X(g)=\{x_1=g, x_2,\dots ,x_{c(g)}\}$ be the set which contains the representatives of the classes of $\overline{G}_n$ coming from the coset $N^*g$. Then for $x_j \in X(g)$, we have

Theorem 2.1

$$\begin{aligned} ( \psi _s \overline{\beta })^{\overline{G}_n}(x_j)=\sum _{y_k:(s,y_k)\in R(g)}\left[ \sum _{l}^{\prime } \frac{|C_{\overline{G}_n} (x_j)|}{|C_{\overline{H_s}} (y_{l_k})|} \psi _s(y_{l_k})\right] \beta (y_k)\end{aligned}$$

Proof

See [16]$\square$

The Fischer-Clifford matrix $M(g) = \left( a_{(s, y_k)}^{j} \right)$ is then defined as

$$\begin{aligned} \left( a_{(s, y_k)}^{j} \right) = \left( \sum _{l}^{\prime } \frac{|C_{\overline{G}_n} (x_j)|}{|C_{\overline{H_s}} (y_{l_k})|} \psi _s(y_{l_k}) \,\right) , \end{aligned}$$

with columns indexed by X(g) and rows indexed by R(g) and where $\sum _{l}^{\prime }$ is the summation over all l for which $y_{l_k}$ is conjugate to $x_j$ in $\overline{G}_n$. In practice it is difficult to compute the elements $y_{l_k}$ or the ordinary irreducible character tables of the inertia groups $\overline{H_s}$, since the sets Irr$(\overline{H_s})$ of ordinary irreducible characters of the $\overline{H_s}$’s are in general much larger and more complicated to compute than the one for $\overline{G}_n$. Instead of using the above formal definition of a Fischer-Clifford matrix M(g), the arithmetical properties of M(g) below are used to compute the entries of M(g) (see [11]).

• (1) $a^j_{(1,g)} = 1$ for all $j = \{1,2,.., c(g)\}$.

• (2) $|X(g)| = |R (g)|$.

• (3) $\sum _{j = 1}^{\widehat{c (g)}} m_j \, a_{(s, y_k)}^{j} \, \overline{ a_{(s^{\prime }, y_{k}^{\prime })}^{j} } = \delta _{(s, y_k),(s^{\prime }, y_{k}^{\prime })} \, \frac{|C_G (g)|}{|C_{{H_s}} (y_k)|} \, |N^*|$.

• (4) $\sum _{(s, y_k) \in R (g)} a_{(s, y_k)}^{j} \, \overline{ a_{(s, y_{k})}^{j^{\prime }}} \, |C_{{H_s}} (y_k)| = \delta _{j j^{\prime }} \, |C_{\overline{G}_n} (x_j)|$.

• (5) M(g) is square and non-singular.

  Since $N^*$ is an elementary abelian 2-group, we have the following additional properties of M(g):

• (6) $a_{(s, y_k)}^{1} = \frac{|C_G (g)|}{|C_{{H_s}} (y_k)|}$.

• (7) $|a_{(s, y_k)}^{1}| \ge |a_{(s, y_k)}^{j}|$.

• (8) $a_{(s, y_k)}^{j}\equiv a_{(s, y_k)}^{1}$(mod 2).

Applying properties (1) and (6), together with the fact that $H_1=H_2$ and $H_3=H_4$, the first column and first row of M(g) are determined. By setting up a system of equations using the row and column orthogonality properties (3) and (4) together with the additional properties (7) and (8) above, the remainder of the entries $a_{(s, y_k)}^{j}$, $j=2,3,\dots ,c(g)$ and $s= 2,3,4$, of M(g) are computed. A Fischer-Clifford matrix M(g) for $\overline{G}_n$ has general structure as depicted in Fig. 1. Note where we write, for example, $\pm a_{(3, y_2)}^{2}$, it does not mean that we obtain two values for this particular entry. It simply means that $\pm a_{(3, y_2)}^{2}$ can only assume one of the two values $a_{(3, y_2)}^{2}$ or $-a_{(3, y_2)}^{2}$ since M(g) is uniquely determined for $\overline{G}_n$.

The Fischer-Clifford matrix M(g) (see Fig. 1) is partitioned row-wise into blocks $M_s(g)$, where each block corresponds to an inertia group $\overline{H_s}$. We write $|C_{\overline{G}_n}(x_j)|$, for each $x_j\in X(g)$, at the top of the columns of M(g) and at the bottom we write $m_j\in \mathbb {N}$. On the left of each row we write $|C_{{H_s}} (y_k)|$, where the conjugacy classes $[y_k]$, $k=1,2,...,r$, of an inertia factor $H_s$ fuse into the conjugacy class [g] of G. Since $|X(g)|=|R(g)|$ it follows that M(g) is a square matrix of size c(g). When there is no class fusion of an inertia factor $H_s$ into a class [g], the block $M_s(g)$ is omitted from M(g).

Fig. 1

Fischer-Clifford Matrix M(g) of $\overline{G}_n$

2.2 The generalized matrix M(1A) of $\overline{G}_n$

In this section, the Fischer-Clifford matrix M(1A) corresponding to the identity class 1A of $Sp_{2n-2}(2)$ is generalized for the family of groups $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)$. Using the facts that $a^{j}_{(1,g)}=1$ for all $j = \{1, 2, \cdots , c(g)\}$, the entries of the first column of M(1A) are equal to the orbit lengths $|\triangle _{1}|$, $|\triangle _{2}|$, $|\triangle _{3}|$ and $|\triangle _{4}|$ of the action of G on Irr$(N^{*})$ and the weights $m_{j}$ equal the orbit sizes $|\Gamma _{1}|$, $|\Gamma _{2}|$, $|\Gamma _{3}|$ and $|\Gamma _{4}|$ of the action of G on $N^{*}$, the matrix M(1A) has the format as depicted in Fig. 2.

Fig. 2

The Fischer-Clifford Matrix M(1A)

By the coset analysis technique, the centralizer orders $|C_{\overline{G}_n}(x)|=\frac{k}{f_j}|C_{G}(g)|$ for the classes $[x_{1}]_{\overline{G}_n}$, $[x_{2}]_{\overline{G}_n}$, $[x_{3}]_{\overline{G}_n}$ and $[x_{4}]_{\overline{G}_n}$ are computed as follows,

$$\begin{aligned}{} & {} |C_{\overline{G}_n}(x_{1})|= \frac{2^{2n-1}}{1}|C_{G}(g)|\\{} & {} |C_{\overline{G}_n}(x_{2})|= \frac{2^{2n-1}}{2^{n-2}(2^{n-1}-1)}|C_{G}(g)|\\{} & {} |C_{\overline{G}_n}(x_{3})|= \frac{2^{2n-1}}{2^{n-2}(2^{n-1}+1)}|C_{G}(g)|\\{} & {} |C_{\overline{G}_n}(x_{4})|= \frac{2^{2n-1}}{2^{2n-2}-1}|C_{G}(g)| \end{aligned}$$

Furthermore, $|a^{1}_{(s,y_{k})}|\ge |a^{j}_{(s,y_{k})}|$, for $j=1,2,\ldots , c(g)$, and since $N=2^{2n-1}$ is an elementary abelian 2-group, $a^{1}_{(s,y_{k})}=a^{j}_{(s,y_{k})}$ (mod 2) for all $j\ge 2$, it follows that $a, b, c\in \{-1,1\}$. First, we let $a=-1$, then using $a+d+g=-1$, $d=-g$ is obtained. Using the column orthogonality relation,

$$\begin{aligned}{} & {} |C_{G}(g)| \times 1^{2}+|C_{G}(g)| \times (-1)^{2}+|C_{H_{3}}(y_{1})| \times (-g)^{2}+|C_{H_{4}}(y_{1})| \times g^{2}\\{} & {} \quad =\frac{2^{2n-1}}{2^{n-2}(2^{n-1}-1)}|C_{G}(g)|. \end{aligned}$$

Since $|C_{H_{3}}(y_{1})|=|C_{H_{4}}(y_{1})|=\frac{|C_{G}(g)|}{2^{2n-2}-1}$ the equation above becomes,

$$\begin{aligned} 1+1+\frac{g^{2}}{2^{2n-2}-1}+\frac{g^{2}}{2^{2n-2}-1}=\frac{2^{2n-1}}{2^{n-2}(2^{n-1}-1)}. \end{aligned}$$

Making g the subject of the formula, gives g and d as follows:

$g=\sqrt{\frac{[\frac{2^{2n-1}}{2^{n-2}(2^{n-1}-1)}-2]2^{2n-2}-1}{2}}$, $d=-\sqrt{\frac{[\frac{2^{2n-1}}{2^{n-2}(2^{n-1}-1)}-2]2^{2n-2}-1}{2}}.$

Proceeding as in the above case by first letting let $b=-1$, e and h are obtained as,

$h=-\sqrt{\frac{[\frac{2^{2n-1}}{2^{n-2}(2^{n-1}+1)}-2]2^{2n-2}-1}{2}}$, $e=\sqrt{\frac{[\frac{2^{2n-1}}{2^{n-2}(2^{n-1}+1)}-2]2^{2n-2}-1}{2}}.$

It is worth noting that for the above two cases, if a or b is taken to be equal to 1 the resulting entries of d and e would violate the condition that $a^{1}_{(s,y_{k})}=a^{j}_{(s,y_{k})}$ mod 2 for all $j \ge 2$. Using the row orthogonality relation for rows 1 and 2 and taking $a=b=-1$, c is determined as follows,

$$\begin{aligned}{} & {} (1 \times 1 \times 1)+(2^{n-2}(2^{n-1}-1) \times -1 \times 1)\\{} & {} +(2^{n-2}(2^{n-1}+1) \times -1 \times 1)+(2^{2n-2}-1) \times c \times 1)=0 \\{} & {} 1-2^{2n-3}+2^{n-2}-2^{2n-3}-2^{n-2}+(2^{2n-2}-1)c=0 \\{} & {} (2^{2n-2}-1)c=2^{2n-2}-1 \\{} & {} c=1 \end{aligned}$$

Using the equation, $c+f+i=-2$, with $c=1$ and by the column orthogonality relation for the last column, we obtain,

$$\begin{aligned}{} & {} |C_{G}(g)| \times (1^{2})+|C_{G}(g)| \times (1)^{2}+\frac{|C_{G}(g)|}{2^{2n-2}-1} \times (f)^{2}\\{} & {} \quad +\frac{|C_{G}(g)|}{2^{2n-2}-1} \times (i^{2})=\frac{2^{2n-1}}{2^{2n-2}-1}|C_{G}(g)| \\{} & {} 2+\frac{f^{2}}{2^{2n-2}-1}+\frac{i^{2}}{2^{2n-2}-1})=\frac{2^{2n-1}}{2^{2n-2}-1} \end{aligned}$$

Multiplying through by $2^{2n-2}-1$ and substituting $i=-2-f$ in the resulting equation,

$$\begin{aligned}{} & {} 2(2^{2n-2}-1)+f^{2}+i^{2}=2^{2n-1} \\{} & {} 2.2^{2n-2}-2+f^{2}+4+4f+f^{2}=2^{2n-1} \\{} & {} 2^{2n-2}[2-2]+2+4f+2f^{2}=0 \\{} & {} f^{2}+2f+1=0 \end{aligned}$$

The resulting quadratic equation gives $f=-1$ and so $i=-2-f=-2+1=-1$. Therefore, the general form of M(1A) for $\overline{G}_n=N^{*}{:}Sp_{2n-2}(2)$ is obtained in Fig. 3.

Fig. 3

The Fischer-Clifford Matrix M(1A)

Using the symbols for orbit lengths and $N^*=2^{2n-1}$ the Fischer-Clifford matrix M(1A) is written as in Fig. 4 below,

Fig. 4

The Fischer-Clifford Matrix M(1A)

2.3 The construction of the character table of $\overline{G}_n$

The partial character table of $\overline{G}_n$ on the classes from the coset $N^*g$ with class representatives in the set X(g), is given by

Fig. 5

Partial Character Table corresponding to $N^*g$

In Fig. 5, $C_s(g)$ indicates the columns of the set Irr$(H_s)$ associated with the classes $[y_k]$, $k=1, 2,...,r$ of an inertia factor $H_s$ which fuse into the class [g]. Whereas $M_s(g)$ is the block of rows of M(g) which is indexed by the centralizer orders $|C_{H_s}(y_k)|$ of the classes $[y_k]$ of $H_s$. To obtain the sub-matrix $[C_s(g)M_s(g)]$ of the partial character table in Fig. 5, we multiply the matrix $[C_s(g)]$ of size $|\text {Irr}(H_s)|\times r$ with the matrix $[M_s(g)]$ of size $r \times |X(g)|$. Furthermore, if there is no class fusion of the inertia factor $H_s$ into [g], then the block $M_s(g)$ corresponding to $H_s$ is omitted from M(g) and therefore the entries of the sub-matrix $[C_s(g)M_s(g)]$ will be all zeroes. The full ordinary character table of $\overline{G}_n$ will have the structure as represented in Fig. 6, where $k(H_1)$ denotes the number of classes of $H_1$.

Fig. 6

Structure of the Character Table of $\overline{G}_n$

Note the set $\{1, g_1, g_2,..., g_{k(H_1)}\}$ are the representatives of conjugacy classes of $H_1$. The ordinary character table of $\overline{G}_n$ in Fig. 6 is a $k(\overline{G}_n) \times k(\overline{G}_n)$ complex-valued square matrix and is partitioned row-wise into four blocks $\Delta _{1}=\{\chi _{i}\vert 1\le i \le k(H_1)\}$, $\Delta _{2}=\{\chi _{i}\vert 1+k(H_1)\le i \le k(H_1)+k(H_2)\}$, $\Delta _{3}=\{\chi _{i}\vert 1+k(H_1)+k(H_2)\le i \le k(H_1)+k(H_2)+k(H_3) \}$ and $\Delta _{4}=\{\chi _{i}\vert 1+k(H_1)+k(H_2)+k(H_3)\le i \le k(H_1)+k(H_2)+k(H_3)+k(H_4) \}$, where $\chi _i \in \text {Irr}(\overline{G}_n)$. Each column $|C_s(g)M_s(g)|, s=1,2,3,4$ of Fig. 6 is indexed by a set X(g) corresponding to a class [g] of $H_1$. The faithful irreducible characters of $\overline{G}_n$ appear in the blocks $\Delta _3$ and $\Delta _4$.

Proposition 2.2

If $n\ge 4$, then the derived subgroup $\overline{G}_n'$ of $\overline{G}_n$ is isomorphic to $\overline{F}_n$

Proof

Note that only the first two rows of M(1A) in Fig. 3, which are associated with $H_1$ and $H_2$, will produce linear characters for $\overline{G}_n$. Since $H_1=H_2$ is simple for $n \ge 4$, $H_1$ has only one linear character. Hence, constructing the partial character table for $\overline{G}_n$ associated with the identity coset N1A we obtain two linear characters of $\overline{G}_n$. It follows that $[\overline{G}_n,\overline{G}_n']=2$. Therefore, $\overline{G}_n'\cong \overline{F}_n=2^{2n-2}{:}Sp_{2n-2}(2)$ since $\overline{F}_n$ is the only subgroup of index 2 in $\overline{G}_n$ up to isomorphism. $\square$

Remark 2.3

From the character table of $\overline{G}_2=2^3{:}Sp(2,2)$ (see Table 5.3.6 in [14]) it can be seen that $\overline{G}_2$ has 4 linear characters and hence $[\overline{G}_2,\overline{G}_2']=4$. Therefore, with the aid of GAP we identified that $\overline{G}_2'\cong A_4$. Also, the ordinary character table of $\overline{G}_3=2^5{:}Sp(4,2)$(see Table F.1 in [17]) contains 4 linear characters. Hence $[\overline{G}_3,\overline{G}_3']=4$ and therefore with the aid of GAP the structure of $\overline{G}_3'$ is identified as $2^4{:}A_6$.

3 An example: The group $\overline{G}_5=2^{9}{:}Sp_{8}(2)$

3.1 Construction of $\overline{G}=2^{9}{:}Sp_{8}(2)$

With the aid of GAP, the group $Sp_{10}(2)$ is generated as matrix group of degree 10 over GF(2). Then the affine subgroup $ASp_5=2^{9}{:}Sp_{8}(2)$ of $Sp_{10}(2)$ is constructed as a matrix group of degree 10 and converted to a permutation group P on 1024 points. Now, using the generators of P the group $ASp_5$ is constructed inside MAGMA. Using the MAGMA subroutine "a:=ChiefSeries(P);","N:= a[2];","M:=GModule(P,N);" and "M:Maximal;" the group $Sp_{8}(2)$ is constructed as a matrix group of degree 9 over GF(2) with 3 generators $g_{1}$, $g_{2}$ and $g_{3}$ (see Fig. 7) of orders 2, 4 and 2, respectively. The group $G=\left\langle g_1^t,g_2^t,g_3^t\right\rangle \cong Sp_{8}(2)$ is generated by the transposes $g_1^t,g_2^t$ and $g_3^t$ of $g_{1}$, $g_{2}$ and $g_{3}$, so that the dual $N^*\cong 2^9$ of $N=2^9$ is a 9-dimensional G-module over GF(2). The group $\overline{G}_5=N^*{:}G\cong 2^9{:}Sp(8,2)$ can be constructed as a matrix group of degree 10 over GF(2), where the determinants of its generators are all equal to 1. Therefore, $\overline{G}_5\le SL_{10}(2)$.

Fig. 7

Generators of $Sp_8(2)$

3.2 The Action of $G=Sp_{8}(2)$ on $2^{9}$

From now on, let $\overline{G}_{5}$ be the split extension of $N^{*} =2^9$ by $G=Sp_{8}(2)$, where G acts as a matrix group of degree 9 on the vector space $N^{*} \cong V_{9}(2)$ of dimension 9 over GF(2). As discussed in Sect. 1, G has 4 orbits of lengths 1, 255, 136 and 120 on the classes of $N^{*}$ with corresponding point stabilizers $P_{1}=Sp_{8}(2)$, $P_{2}=2^{7}{:}Sp_{6}(2)$, $P_{3}=O^{+}_{8}(2){:}2$ and $P_{4}=O^{-}_{8}(2){:}2$, respectively. Checking the indices of the maximal subgroups of G in the ATLAS [5], it is obvious that $P_{2}$, $P_{3}$ and $P_{4}$ sit maximally inside G.

3.3 Permutation Character of G on $N^*$

In the ATLAS, the permutation characters of G on the classes of $P_s$, $s=1,2,3,4$, are given as $\chi \left( G|P_1\right) =1a$, $\chi \left( G|P_2\right) =1a+119a+135a$, $\chi \left( G|P_3\right) =1a+135a$ and $\chi \left( G|P_4\right) =1a+119a$. It follows that the permutation character of G on the classes of $N^*$ is written as $\chi (G \vert 2^{9})=\sum _{s=1}^4{\chi \left( G|P_s\right) }=1aaaa+119aa+135aa$. The value $k=\chi (G \vert N)(g)=2^{n}$, $n \in \{0,1,2, \cdots ,9\}$, gives the number k of elements of $N^*$ fixed by $g\in G$. The k-value is constant on a class [g] of G and these k-values are listed in Table 1.

Table 1 Permutation Character of $G=Sp_{8}(2)$ on $2^{9}$

3.4 The Conjugacy Classes of $\overline{G}_5=2^{9}{:}Sp_{8}(2)$

The conjugacy classes of $\overline{G}_5$ are computed using the coset analysis technique (see [10, 11]). For each class representative $g\in G$, the classes of $\overline{G}_5$ are computed from a coset $N^*g$ of $N^*$ in $\overline{G}_5$. The formula $|C_{\overline{G}_5}(x_j)|=\frac{k}{f_j}|C_{G}(g)|$ is used to compute the centralizer order $|C_{\overline{G}_5}(x_j)|$ of a class representative $x_j\in \overline{G}_5$ in the set $X(g)=\left\{ x_1,x_2\ldots ,x_{c(g)}\right\}$ coming from $N^*g$. The k-value obtained in Sect. 3.3 is the number of orbits $Q_{1}$, $Q_{2}$, $\cdots ,$ $Q_{k}$ of the action by conjugation of $N^*$ on $N^*g$. Under the action of $C_{G}(g)$, $f_j$ of the orbits $Q_{1}$, $Q_{2}$, $\dots ,$ $Q_{k}$ fuse together to form one orbit $O_{f_j}$. To compute the $f_j$’s and the orders of class representatives $x_j \in X(g)$, the Programme A in [3] written in GAP is used. The constant $m_{j}=\frac{f_j}{k}|N|$ corresponding to each class $[x_j]$ of $\overline{G}_5$ is also computed and is used in the construction of a Fischer-Clifford matrix M(g) of $\overline{G}_5$. By the technique of coset analysis, it is found that $\overline{G}_5$ has a total of 390 conjugacy classes (see Table 2). Since $SL_{10}(2)$ is a very large group, our available computer devices could not complete the computation of the ordinary character table of $SL_{10}(2)$ in GAP. Therefore, the computation of the fusion map of the classes of $\overline{G}_5$ into $SL_{10}(2)$ using their character tables could not be carried out. To resolve this, the fusion of conjugacy classes of $\overline{G}_5$ (represented as a permutation on 496 points) into $SL_{10}(2)$ was first carried out followed by the fusion of classes of the character table constructed in Sect. 3.7 (denoted by ct) into the character table of $\overline{G}_5$ which was directly computed by GAP. The two fusions are linked and the resulting fusion map of $\overline{G}_5$ into $SL_{10}(2)$ is listed in the last column of Table 2. The GAP subroutine below was used to assist us in the process of computing the fusion map of $\overline{G}_5$ into $SL_{10}(2)$.

gap> P:=SL(10,2);;V:=FullRowSpace(GF(2), 9);;

gap> S:=SemidirectProduct(G,V);

gap> CL:=ClassNames(CharacterTable(P));

gap> CCL:=ClassNames(CharacterTable(S));

gap> F1:=FusionConjugacyClasses(S,P);

gap> SieveF1Classes:=CL{F1};

gap> F2:=PossibleClassFusions(ct,CharacterTable(S));

gap> SieveF2Classes:=CCL{F2[1]};

Table 2 The Conjugacy Classes of $\overline{G}_5=2^{9}{:}Sp_{8}(2)$

3.5 Inertia Factor Groups of $\overline{G}_5$

From Sect. 1, the action of G on Irr$(N^*)$ gives four orbits of lengths 1, 1, 255 and 255 with corresponding inertia factor groups $H_{1}=H_{2}=Sp_{8}(2)$, $H_{3}=H_{4}=2^{7}{:}Sp_{6}(2)$. The fusion map of $H_3$ into $Sp_{8}(2)$ can be obtained from Table 6.9 in [1] and is listed in Table 3.

Table 3 The fusion of $H_{3}=H_{4}$ into $G = Sp_{8}(2)$

3.6 The Fischer-Clifford Matrices of $\overline{G}_5$

The Fischer-Clifford matrices of $\overline{G}_5$ are computed by using their properties in Sect. 2.1. Sometimes, generating the equations satisfied by the entries and capturing these equations in a Microsoft Excel Worksheet also aided in narrowing down to the correct entries and at the same time checking whether all the equations are satisfied. For instance, in a Microsoft Excel Worksheet one can easily check that the column orthogonal relation results to a constant whenever applied on any two different columns. This is illustrated by the partial Fischer-Clifford matrix M(4J) in Fig. 8. The desired Fischer-Clifford matrices are obtained by appropriately swapping the rows and columns of the resulting matrix. By the fact that $G=H_{2}$ and $H_{3}=H_{4}$, the sizes of the Fischer-Clifford matrices of $\overline{G}_5$ are even. The Fischer-Clifford matrices are displayed in Table 4.

Fig. 8

The Fischer-Clifford matrix M(4J) on Excel

Table 4 The Fischer-Clifford Matrices of $\overline{G}_5 = 2^{9}{:}Sp_{8}(2)$

3.7 The Character Table of $\overline{G}_5 = 2^{9}{:}Sp_{8}(2)$

Using the fusion map of the classes of $H_{3}=H_{4}=2^{7}{:}Sp_{6}(2)$ into G, the ordinary character tables of $Sp_{8}(2)$ and $H_3$ found in the GAP library and the Fischer-Clifford matrices of $\overline{G}_5$, the ordinary character table of $\overline{G}_5$ is constructed by the Fischer-Clifford matrices technique as outlined in Sect. 2.3. The ordinary character table of $\overline{G}_5$ is a $390 \times 390$ complex valued matrix. The irreducible characters $\chi _i$ of $\overline{G}_5$ are partitioned into blocks, $B_{1}=\{\chi _{i}\vert 1\le i \le 81\}$, $B_{2}=\{\chi _{i}\vert 82 \le i \le 162\}$, $B_{3}=\{\chi _{i}\vert 163\le i \le 276 \}$ and $B_{4}=\{\chi _{i}\vert 277 \le i \le 390 \}$ which are associated with each inertia group $\overline{H}_s$, $i=1,2,3,4$. The consistency and accuracy of the character table of $\overline{G}_5$ has been tested using the GAP routine, Programme E by [15], and is accessible via the second link below. The complete ordinary character table of $\overline{G}_5$ in pdf-format is accessible via the first link below. Using Programme E, unique power maps for the classes of $\overline{G}_5$ are obtained and are listed in Table 2.

https://drive.google.com/file/d/1aw80YL4-Fb2Hm8_8W4Py7O4kAV0S0fDx/view?usp=drive_link

https://drive.google.com/file/d/1opwb_V5lJURkBqNAApYmcGPng2fNbxc2/view?usp=sharing