Subgroup ($H$) information
| Description: | $C_3^2:D_6\times S_4$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(10,17,13)(12,16,14), (1,8,9)(2,3,5)(4,6,7), (13,17)(14,16), (1,6,5)(2,8,7) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_6^2:C_2^2.S_4^2$ |
| Order: | \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^9.A_4\wr C_3$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $S_4\times \AGL(2,3).C_2^2$ |
| $W$ | $S_4\times S_3^2$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^2:D_6^2$ |
| Normal closure: | $C_6^2:C_2^2.S_4^2$ |
| Core: | $C_2\times C_6^2$ |
Other information
| Number of subgroups in this autjugacy class | $48$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | not computed |
| Projective image | not computed |