Subgroup ($H$) information
| Description: | $C_6^2:D_6$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,9)(5,6)(7,8)(10,12)(15,16)(17,19)(18,21)(20,22), (16,18,17), (15,19,21) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^5:(C_2^2\times S_4)$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_6^2).C_3^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $D_6^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_2^2:S_3^3$ | ||
| Normal closure: | $C_3^5:(C_2\times S_4)$ | ||
| Core: | $C_2^2$ | ||
| Minimal over-subgroups: | $C_6^2:S_3^2$ | $C_6^2:S_3^2$ | $C_2^2:S_3^3$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | $C_3^5:(C_2^2\times S_4)$ |