Subgroup ($H$) information
| Description: | $C_2.\SL(2,7).\SO(3,7)$ |
| Order: | \(225792\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7^{2} \) |
| Index: | \(2\) |
| Exponent: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rrrr}
6 & 0 & 0 & 0 \\
0 & 6 & 0 & 0 \\
0 & 0 & 6 & 0 \\
0 & 0 & 0 & 6
\end{array}\right), \left(\begin{array}{rrrr}
2 & 3 & 6 & 0 \\
6 & 5 & 6 & 6 \\
2 & 0 & 2 & 4 \\
5 & 2 & 1 & 5
\end{array}\right), \left(\begin{array}{rrrr}
5 & 5 & 1 & 4 \\
3 & 0 & 0 & 1 \\
0 & 0 & 5 & 2 \\
0 & 0 & 4 & 0
\end{array}\right), \left(\begin{array}{rrrr}
1 & 5 & 4 & 0 \\
4 & 6 & 0 & 3 \\
2 & 0 & 6 & 5 \\
0 & 5 & 4 & 1
\end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, and nonsolvable. Whether it is a direct factor, a semidirect factor, or almost simple has not been computed.
Ambient group ($G$) information
| Description: | $\SL(2,7)^2.C_2^2$ |
| Order: | \(451584\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7^{2} \) |
| Exponent: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.\PSL(2,7)^2.D_4$ |
| $\operatorname{Aut}(H)$ | $C_2^2.\PSL(2,7)^2.D_4$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |