Subgroup ($H$) information
| Description: | $C_6.S_3^2$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a, g, f, b^{2}c^{2}d^{2}e^{2}g^{2}, c^{3}df, c^{2}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and rational.
Ambient group ($G$) information
| Description: | $C_3^3:S_3^2:C_2^2$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_3^2.C_4.C_2^4$, of order \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $(C_2^2\times \He_3):D_4$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $\He_3:D_4$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_3^3:S_3^2:C_2^2$ |