Subgroup ($H$) information
| Description: | $C_2^6:A_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(3,4)(7,8)(19,20)(23,24), (3,4)(5,6)(9,10)(11,12)(19,20)(23,24), (13,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2^8.S_4$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3.C_2^2$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $(C_2^5\times C_6).C_6.\PSL(3,4).S_3$, of order \(139345920\)\(\medspace = 2^{14} \cdot 3^{5} \cdot 5 \cdot 7 \) |
| $W$ | $C_2^4:S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6:S_4$ |
| Normal closure: | $C_2^8.A_4$ |
| Core: | $C_2^8$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |