Subgroup ($H$) information
| Description: | $C_3^6:A_4\wr C_2$ |
| Order: | \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(2,9,3)(4,6,13)(16,17,18), (1,5,14)(11,12,15)(16,17,18), (16,17,18), (1,5,14) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.A_4^2:D_4$ |
| Order: | \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| 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 \(10077696\)\(\medspace = 2^{9} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.A_4^2:C_2^3$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $W$ | $C_3^6.A_4^2:C_2^2$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^7.A_4^2:C_2^2$ |
| Normal closure: | $C_3^7.A_4^2:C_2^2$ |
| Core: | $C_3^6:(C_2^2:A_4)$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^7.A_4^2:D_4$ |