Subgroup ($H$) information
| Description: | $C_3:Q_8$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ab, c^{33}, c^{66}, c^{88}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{12}.D_{22}$ |
| Order: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_3\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_6:Q_8$ | |||
| Normal closure: | $C_{33}:Q_8$ | |||
| Core: | $C_{12}$ | |||
| Minimal over-subgroups: | $C_{33}:Q_8$ | $C_6:Q_8$ | ||
| Maximal under-subgroups: | $C_{12}$ | $C_3:C_4$ | $C_3:C_4$ | $Q_8$ |
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $1$ |
| Projective image | $S_3\times D_{22}$ |