Subgroup ($H$) information
| Description: | $S_3\times A_4^2$ |
| Order: | \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a^{2}b^{3}de^{2}f^{2}g^{4}, e^{3}, f^{3}g^{3}, cde^{2}f^{4}g^{4}, d^{3}e^{3}, f^{2}g^{2}, g^{3}, d^{2}e^{4}f^{5}g^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $S_4^2:D_6$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| $W$ | $A_4^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $A_4^2:D_6$ | |||
| Normal closure: | $C_6^4.C_3.C_6$ | |||
| Core: | $C_2^4$ | |||
| Minimal over-subgroups: | $C_3:S_3\times A_4^2$ | $C_3^2.A_4^2.C_2$ | $C_3^2.A_4^2.C_2$ | $A_4^2:D_6$ |
Other information
| Number of subgroups in this autjugacy class | $54$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:(C_2\times A_4^2:C_4)$ |