Subgroup ($H$) information
| Description: | $A_4^2:S_3$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$b^{3}, d^{3}e^{3}, e^{3}, cd^{4}e^{5}f^{4}, f^{2}g^{2}, g^{3}, f^{3}, d^{2}e^{4}f^{2}g^{3}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
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 \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $(C_3\times A_4^2).C_3:S_3.C_2^3$ |
| $\card{W}$ | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $(C_3\times A_4^2).C_6$ | |
| Normal closure: | $A_4^2.C_3:S_3$ | |
| Core: | $C_2^4$ | |
| Minimal over-subgroups: | $A_4^2.C_3:S_3$ | $(C_3\times A_4^2).C_6$ |
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:(C_2\times A_4^2:C_4)$ |