Subgroup ($H$) information
| Description: | $C_3:S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
49 & 69 \\
48 & 7
\end{array}\right), \left(\begin{array}{rr}
64 & 21 \\
63 & 43
\end{array}\right), \left(\begin{array}{rr}
49 & 36 \\
48 & 13
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $C_2^3.S_3^3$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_2^2.D_6^2\times S_3$ |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $S_3^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_3:S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $D_6$ | ||||
| Normalizer: | $C_6:S_3^2$ | ||||
| Normal closure: | $C_6:S_4$ | ||||
| Core: | $C_3$ | ||||
| Minimal over-subgroups: | $C_3:S_4$ | $C_3^2:C_6$ | $C_6:S_3$ | $C_6:S_3$ | $C_6:S_3$ |
| Maximal under-subgroups: | $C_3^2$ | $S_3$ | $S_3$ | $S_3$ | $S_3$ |
Other information
| Number of subgroups in this conjugacy class | $8$ |
| Möbius function | $0$ |
| Projective image | $C_2^3.S_3^3$ |