Subgroup ($H$) information
| Description: | $C_3\times S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rrr}
1 & 0 & 0 \\
1 & 1 & 0 \\
0 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 1
\end{array}\right), \left(\begin{array}{rrr}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 1
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_3^2:\GL(2,3)$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 24T1325.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3\) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3$ | ||
| Normalizer: | $S_3^2$ | ||
| Normal closure: | $\PU(3,2)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^2:C_6$ | $S_3^2$ | |
| Maximal under-subgroups: | $C_3^2$ | $C_6$ | $S_3$ |
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_3^2:\GL(2,3)$ |