Subgroup ($H$) information
| Description: | $C_2^5:\GL(3,2)$ |
| Order: | \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| Index: | \(2187\)\(\medspace = 3^{7} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(22,23), (5,20)(8,14)(11,18)(12,19)(22,23), (1,16)(2,12,10,19)(5,6,20,17) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3^7.C_2^5.\PSL(2,7)$ |
| Order: | \(11757312\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 7 \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.C_2^5.\PSL(2,7)$, of order \(11757312\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^4.\PSL(2,7)\times S_3$ |
| $W$ | $C_2^3:\GL(3,2)$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^5:\GL(3,2)$ |
| Normal closure: | $C_3^7.C_2^5.\PSL(2,7)$ |
| Core: | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2187$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.C_2^4:\GL(3,2)$ |