Subgroup ($H$) information
| Description: | $C_6^2:S_3^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| 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: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
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 \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $\He_3.C_6^2.C_2^5$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| $W$ | $(C_6\times \He_3):D_4$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed | |
| Normalizer: | $\He_3.C_2^4:S_3$ | |
| Normal closure: | $C_3^2.C_3^4.C_2^4:S_3$ | |
| Core: | $C_3$ | |
| Minimal over-subgroups: | $C_3^4.D_6.C_2^2$ | $\He_3.C_2^4:S_3$ |
Other information
| Number of subgroups in this autjugacy class | $54$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^3.S_3\wr C_2^2$ |