Subgroup ($H$) information
| Description: | $C_2\times C_6^4.S_3^2$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,12,18)(3,14,16), (1,7)(2,8,5,4,9,13)(3,10,16,15,14,6)(11,18)(12,17)(19,25,28,27) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_6^5.C_2.S_5$ |
| Order: | \(1866240\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^5.C_2^2.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2^8.D_5^2.C_2^3$, of order \(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \) |
| $W$ | $C_6^4:(C_2\times D_6)$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.D_6^2$ |
| Normal closure: | $C_6^5.S_5$ |
| Core: | $C_6^5$ |
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |