Subgroup ($H$) information
| Description: | $(C_3\times C_{24}):C_4$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(7,13,9,8,12,15,10,11), (7,12)(8,11)(9,10)(13,15), (2,5,6)(7,12)(8,11)(9,10) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^4:(D_4\times \GL(2,3))$ |
| Order: | \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:S_3.C_6^2.C_6.D_4.C_2$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_4\times C_3:S_3.C_2).C_2^5$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| $W$ | $C_3^2:D_4^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^4:(D_4\times \GL(2,3))$ |