Subgroup ($H$) information
| Description: | $A_4:Q_8$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(7\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, d, cd, b^{85}, b^{154}, b^{105}$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and a $96$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial).
Ambient group ($G$) information
| Description: | $C_{28}.S_4$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times S_4\times F_7$, of order \(8064\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $-1$ |
| Projective image | $C_{14}:S_4$ |