Subgroup ($H$) information
| Description: | not computed |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$a^{3}, e^{3}, d^{2}e^{4}f^{2}g^{3}, f^{2}g^{2}, f^{3}, b^{3}, cde^{2}f^{4}g^{4}, d^{3}, g^{2}, g^{3}, a^{2}d^{3}e^{3}fg^{5}$
|
| Derived length: | not computed |
The subgroup is maximal, nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational 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 \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4:(C_2\times A_4^2:C_4)$ |