Subgroup ($H$) information
| Description: | not computed |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(2,12,6)(5,11,14)(9,18,13), (3,7)(10,17)(11,14)(13,18), (6,12)(9,13)(11,14) \!\cdots\! \rangle$
|
| 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: | $C_3^6.(C_2^7.D_6)$ |
| Order: | \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2^6.C_6.C_2^5$, of order \(8957952\)\(\medspace = 2^{12} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.C_2^5.C_2^4$ |
| Normal closure: | $C_3^6.(C_2^7.D_6)$ |
| Core: | $C_2\times C_3^5.D_6$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_2^5:S_4$ |