Subgroup ($H$) information
| Description: | $C_4^3.(C_2\times A_4)$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(17,18)(19,20)(21,22)(23,24), (9,13,10,14)(11,15,12,16)(17,22,18,21)(19,23,20,24) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_4^3.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5:F_5^2$, of order \(196608\)\(\medspace = 2^{16} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3).C_2^6$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| $W$ | $C_4^2.(C_2^3\times A_4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5:D_5^3$ |