Subgroup ($H$) information
| Description: | $(S_3\times C_6^2):S_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{3}, fg, c^{4}d^{5}e^{4}fg, d^{2}e^{4}, g, b^{3}, b^{2}c^{8}d^{4}e^{5}, e^{3}g, d^{3}, a^{2}g$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_6^3.(D_6\times S_4)$ |
| Order: | \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_5\wr C_4$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_2^2\times C_6^2).D_6^2$ |
| $W$ | $(S_3\times C_6^2):S_4$, 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: | $C_2\times C_6^4:D_6$ |
| Core: | $C_2^4.C_3^3:S_3$ |
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 | $C_6^3.(D_6\times S_4)$ |