Subgroup ($H$) information
| Description: | $C_2^4\times C_6$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(9,11,10), (1,5)(3,4), (12,13)(14,15), (2,6)(7,8), (1,4)(2,6)(3,5)(7,8), (1,4)(2,7)(3,5)(6,8)(12,14)(13,15)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $(C_2^3\times C_6):S_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^5\times C_6).C_3^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times \GL(5,2)$, of order \(19998720\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 31 \) |
| $\operatorname{res}(S)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $(C_2^2\times C_6):S_4$ |