Subgroup ($H$) information
| Description: | $C_6^2.S_4$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(14,16)(15,17), (10,13)(11,12), (3,7,5)(6,9,8), (10,11)(12,13), (1,3,6,2,5,9,4,7,8) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $D_4\times C_6^2.S_3^2$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_6^2).C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^2.S_4\times S_4$ |
| $W$ | $C_6^2.D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_6^2.D_6^2$ |