Subgroup ($H$) information
| Description: | $Q_8\times S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ac^{21}, de^{3}, c^{8}, b, e^{3}, c^{18}, c^{12}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $Q_8:S_3\times S_4$ |
| Order: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.D_6^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_4^2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $\GL(2,\mathbb{Z}/4):C_2^2$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $3$ |
| Projective image | $C_2^4:S_3^2$ |