Subgroup ($H$) information
| Description: | $C_3^4:S_3^2:D_6$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,3)(5,6)(8,9)(10,11)(13,25)(14,27)(15,26)(16,28)(17,30)(18,29)(20,21) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^3.S_3\wr C_2^2$ |
| Order: | \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^5.C_2^3.C_2^4$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^2.C_3^5.C_2.C_2^5$, of order \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
| $W$ | $\He_3^2:D_4:D_6$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
Related subgroups
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 | $C_3^3.S_3\wr C_2^2$ |