Subgroup ($H$) information
| Description: | $C_3^3:S_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$b^{3}f^{3}g^{3}, d^{3}e^{3}, g^{2}, b^{2}defg^{3}, e^{3}, cd^{3}e^{3}fg, f^{2}g^{2}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^4:(C_2\times A_4^2:C_4)$ |
| Order: | \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^4.D_6\wr C_2$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $(C_3\times C_6^2).S_3^3$ |
| $\card{W}$ | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $C_6^2.C_3^3.C_2^2$ | |||
| Normal closure: | $C_2^4.C_3^4.C_3.C_6$ | |||
| Core: | $C_3^2$ | |||
| Minimal over-subgroups: | $C_2^4.C_3^3:S_3$ | $C_3^4:S_4$ | $C_3^4:S_4$ | $C_6^2:S_3^2$ |
Other information
| Number of subgroups in this autjugacy class | $48$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^4:(C_2\times A_4^2:C_4)$ |