Subgroup ($H$) information
| Description: | $C_3:C_8\times S_4$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
1 & 15 \\
10 & 1
\end{array}\right), \left(\begin{array}{rr}
16 & 15 \\
5 & 11
\end{array}\right), \left(\begin{array}{rr}
13 & 4 \\
8 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 8 \\
12 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
11 & 0 \\
10 & 11
\end{array}\right), \left(\begin{array}{rr}
11 & 10 \\
0 & 11
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right)$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_6\times S_4):\OD_{16}$ |
| Order: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times A_4).C_2^6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2^3:D_6^2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| $\card{W}$ | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |