Subgroup ($H$) information
| Description: | $C_3^4:(C_2\times A_4^2:C_4)$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(4,13)(5,6)(8,10)(11,12)(15,18)(16,17)(19,21)(20,24)(22,26)(23,25)(27,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.D_6^2:C_2^2$ |
| Order: | \(746496\)\(\medspace = 2^{10} \cdot 3^{6} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.A_4^2.C_3^2.C_2^5.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_6^4.D_6\wr C_2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $W$ | $C_3^4.S_4^2:C_2^2$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.C_6^2:D_4$ |
| Normal closure: | $C_6^4.C_6^2:C_4$ |
| Core: | $(C_6^2\times A_4).C_3^2.D_6$ |
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 | $C_6^4.D_6^2:C_2^2$ |