Subgroup ($H$) information
| Description: | $C_2\times C_3^4:C_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,3)(5,8)(11,14)(12,13), (1,5,8)(9,12,13)(10,11,14), (1,3)(2,5,7,8)(4,6) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2^5:(C_3^4:C_4)$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.C_3^2.C_4:\SD_{16}.C_2$ |
| $\operatorname{Aut}(H)$ | $C_3^4.C_8.C_2.A_6.C_2^2$ |
| $W$ | $C_3^4:C_4$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $(C_3^2\times A_4^2):C_4$ |