Subgroup ($H$) information
| Description: | $C_3:C_{12}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
4 & 8 \\
8 & 5
\end{array}\right), \left(\begin{array}{rr}
4 & 6 \\
6 & 7
\end{array}\right), \left(\begin{array}{rr}
8 & 0 \\
0 & 8
\end{array}\right), \left(\begin{array}{rr}
4 & 3 \\
3 & 7
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_3^2:D_{12}$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2\times D_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_6$ | ||
| Normalizer: | $C_3:C_{12}$ | ||
| Normal closure: | $C_3^2:C_{12}$ | ||
| Core: | $C_6$ | ||
| Minimal over-subgroups: | $C_3^2:C_{12}$ | ||
| Maximal under-subgroups: | $C_3\times C_6$ | $C_{12}$ | $C_3:C_4$ |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $C_3:S_3^2$ |