Subgroup ($H$) information
| Description: | $C_2^3:\GL(2,\mathbb{Z}/4)$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,17,2,18)(3,19,4,20)(5,16)(6,15)(7,13)(8,14)(9,12)(10,11)(21,23)(22,24) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^7:A_5$ |
| Order: | \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.S_5$ |
| $\operatorname{Aut}(H)$ | $A_4^2.C_2^5.C_2^2$ |
| $W$ | $C_2^3:S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2^6.A_5$ |