Subgroup ($H$) information
| Description: | not computed |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(5,18)(7,17)(8,10)(11,13)(12,15)(14,16)(19,26), (3,4)(5,17)(6,9)(7,18)(11,15) \!\cdots\! \rangle$
|
| 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 |
| $W$ | $C_3^4:(C_2\times A_4^2:C_4)$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.(C_2\times C_6^2:C_4)$ |
| Normal closure: | $C_6^4.C_3^3.C_2^4$ |
| Core: | $C_6^4.C_6^2.C_2$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.(A_4^2:C_4\times D_6)$ |