Subgroup ($H$) information
| Description: | $C_3:D_{12}$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, c^{2}, b^{3}c^{3}d^{3}, d^{3}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^2:D_6$ |
| Order: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $\operatorname{Aut}(H)$ | $D_6^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $S_3\times D_6$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $S_3\times D_6$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $D_6:D_6$ | ||||
| Normal closure: | $C_6^2:D_6$ | ||||
| Core: | $C_3^2$ | ||||
| Minimal over-subgroups: | $D_6:D_6$ | ||||
| Maximal under-subgroups: | $C_6\times S_3$ | $C_6:S_3$ | $C_3:C_{12}$ | $C_3:D_4$ | $D_{12}$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_6^2:D_6$ |