Subgroup ($H$) information
| Description: | $\He_3:S_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{2}d^{3}fg^{5}, f^{3}g^{3}, f^{2}g^{2}, b^{2}defg^{3}, g^{3}, cde^{5}f^{4}g^{4}, d^{2}e^{4}f^{5}g^{3}$
|
| 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 \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $S_4\times C_3^2:\GL(2,3)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $W$ | $A_4:S_3^2$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||
| Normalizer: | $(C_3^3\times A_4).D_6$ | ||
| Normal closure: | $(C_2\times C_6^3).C_3^3.C_2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_2^4.\He_3:S_3$ | $C_3\times \He_3:S_4$ | $C_6^2:S_3^2$ |
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:(C_2\times A_4^2:C_4)$ |