Subgroup ($H$) information
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrrr}
0 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 \\
2 & 0 & 1 & 2 \\
1 & 0 & 0 & 2
\end{array}\right), \left(\begin{array}{rrrr}
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
2 & 0 & 1 & 1 \\
1 & 0 & 0 & 0
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Ambient group ($G$) information
| Description: | $C_3^2:D_6^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times \He_3).C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||||
| Normalizer: | $C_2\times D_6$ | ||||
| Normal closure: | $C_3^2:S_3$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_3:S_3$ | $C_3:S_3$ | $C_3:S_3$ | $D_6$ | $D_6$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $216$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3^2:D_6^2$ |