Subgroup ($H$) information
| Description: | not computed |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | not computed |
| Generators: |
$c^{3}, a^{2}b^{3}c^{2}e^{5}f^{5}g^{5}, g^{3}, c^{2}d^{4}e^{2}f^{5}g^{4}, e^{2}g^{2}, f^{2}g^{4}, e^{3}g^{3}, b^{2}d^{3}e^{2}g^{4}, f^{3}g^{3}, d^{2}e^{5}g^{2}, d^{3}f^{3}, g^{2}$
|
| Derived length: | not computed |
The subgroup is 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: | $C_3^4.S_4^2:C_4$ |
| Order: | \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
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 |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^4.C_3^4:C_3.D_6.C_2$ |
| Normal closure: | $C_2^4.C_3^4:C_3.D_6.C_2$ |
| Core: | $C_2^4.C_3^4:C_3.S_3$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.S_4^2:C_4$ |