Subgroup ($H$) information
| Description: | $C_2^3\times A_4$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(945\)\(\medspace = 3^{3} \cdot 5 \cdot 7 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(11,12)(13,14), (1,4)(3,6)(5,7)(8,9), (1,8)(3,5)(4,9)(6,7), (1,3,8)(4,6,9)(11,14)(12,13), (1,3)(4,6)(5,8)(7,9)(11,14)(12,13), (11,13)(12,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_5\times {}^2G(2,3)$ |
| Order: | \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(630\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times {}^2G(2,3)$, of order \(181440\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_4\times \GL(3,2)$, of order \(4032\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7 \) |
| $W$ | $C_3\times A_4$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $315$ |
| Möbius function | $0$ |
| Projective image | $A_5\times {}^2G(2,3)$ |