Subgroup ($H$) information
| Description: | $C_3^6.A_4^2:D_6$ |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(11,14)(13,17), (4,9)(10,15)(11,14)(13,17), (5,10,15), (2,9,4), (2,9,4) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $S_3\times C_3^6.A_4^2:D_4$ |
| Order: | \(5038848\)\(\medspace = 2^{8} \cdot 3^{9} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2^6:S_3^3$, of order \(10077696\)\(\medspace = 2^{9} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.C_2^5:S_3^3$, of order \(5038848\)\(\medspace = 2^{8} \cdot 3^{9} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^7.A_4^2:C_2^3$ |
| Normal closure: | $C_3^7.A_4^2:C_2^3$ |
| Core: | $C_3^7.A_4^2.C_2$ |
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 | not computed |