Subgroup ($H$) information
| Description: | $(S_3\times C_6^2):S_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{3}c^{3}, f^{3}g, d^{2}f^{2}, d^{2}e^{2}f^{4}, g, b^{3}, c^{2}d^{4}ef^{3}g, e^{3}f^{3}, d^{3}f^{3}, a^{2}d^{3}g$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2\times C_6^4.S_3^2$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_6^3).C_3^4.C_2^6$ |
| $\operatorname{Aut}(H)$ | $(C_2^2\times C_6^2).D_6^2$ |
| $W$ | $(S_3\times C_6^2):S_4$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $(D_6\times C_6^2):S_4$ |
| Normal closure: | $C_6^4.S_3^2$ |
| Core: | $C_2\times C_6^3$ |
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $C_2\times C_6^4.S_3^2$ |