Subgroup ($H$) information
| Description: | $C_{12}.C_2^4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
11 & 5 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
17 & 0 \\
0 & 17
\end{array}\right), \left(\begin{array}{rr}
11 & 0 \\
0 & 11
\end{array}\right), \left(\begin{array}{rr}
13 & 4 \\
8 & 1
\end{array}\right), \left(\begin{array}{rr}
15 & 12 \\
16 & 15
\end{array}\right), \left(\begin{array}{rr}
11 & 10 \\
0 & 11
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_6:C_8\times S_4$ |
| 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).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times A_4).C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $D_6.C_2^7:\GL(3,2)$, of order \(258048\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^6:D_6$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $S_3\times S_4$ |