Subgroup ($H$) information
| Description: | $A_4:Q_8$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(3\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, d^{3}, cd^{3}, b^{3}, b^{8}, b^{6}$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_2^3.S_3^2$ |
| Order: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2:D_6^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2$ | ||||
| Normalizer: | $A_4:Q_8$ | ||||
| Normal closure: | $C_2^3.S_3^2$ | ||||
| Core: | $A_4:C_4$ | ||||
| Minimal over-subgroups: | $C_2^3.S_3^2$ | ||||
| Maximal under-subgroups: | $A_4:C_4$ | $C_4\times A_4$ | $A_4:C_4$ | $C_2^2:Q_8$ | $C_3:Q_8$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $S_3\times S_4$ |