Subgroup ($H$) information
| Description: | not computed |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(19,26)(23,24), (25,30), (22,27)(25,30), (3,4,15,11,12,13)(5,17,18,7,6,9) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.(C_6^2:C_4\times S_3)$ |
| Order: | \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(26873856\)\(\medspace = 2^{12} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.D_6^2$ |
| Normal closure: | $C_6^4.C_3^3.C_2^4$ |
| Core: | $C_2^6\times C_3^2:(C_3^3:C_2)$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.(A_4^2:C_4\times D_6)$ |