Subgroup ($H$) information
| Description: | $C_8.S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | $1$ |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
6 & 0 \\
2 & 10
\end{array}\right), \left(\begin{array}{rr}
2 & 0 \\
15 & 15
\end{array}\right), \left(\begin{array}{rr}
16 & 0 \\
0 & 16
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
4 & 4
\end{array}\right), \left(\begin{array}{rr}
5 & 13 \\
1 & 13
\end{array}\right), \left(\begin{array}{rr}
10 & 3 \\
6 & 7
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
16 & 16
\end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_8.S_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_8$ | ||
| Normalizer: | $C_8.S_4$ | ||
| Complements: | $C_1$ | ||
| Maximal under-subgroups: | $C_8.A_4$ | $\OD_{32}:C_2$ | $C_3:C_{16}$ |
Other information
| Möbius function | $1$ |
| Projective image | $S_4$ |