Subgroup ($H$) information
| Description: | $C_3^5.C_2\wr S_3^2$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,6,12)(2,7,13)(3,8,14)(4,9,15)(5,10,16)(11,17,18), (21,26)(22,27), (22,27) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^6.C_3^5:S_6$ |
| Order: | \(11197440\)\(\medspace = 2^{10} \cdot 3^{7} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^5.C_2^3.A_6.C_2$ |
| $\operatorname{Aut}(H)$ | $(C_2\times C_6^3).C_3^2.C_6^3.C_2^4$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^5.S_4^2:D_4$ |
| Normal closure: | $C_2^6.C_3^5:S_6$ |
| Core: | $C_2\times C_6^5$ |
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 |