Subgroup ($H$) information
| Description: | $C_{16}.D_{12}$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
0 & 1 \\
1 & 0
\end{array}\right), \left(\begin{array}{rr}
130 & 0 \\
0 & 49
\end{array}\right), \left(\begin{array}{rr}
108 & 0 \\
0 & 84
\end{array}\right), \left(\begin{array}{rr}
56 & 0 \\
0 & 139
\end{array}\right), \left(\begin{array}{rr}
109 & 0 \\
0 & 85
\end{array}\right), \left(\begin{array}{rr}
55 & 0 \\
0 & 181
\end{array}\right), \left(\begin{array}{rr}
48 & 0 \\
0 & 21
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 192
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{32}.D_{24}$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:((C_4\times C_8).C_2^6)$ |
| $\operatorname{Aut}(H)$ | $C_3:((C_8\times D_4).C_2^2)$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_3 \rtimes (C_4^2.C_2^4)$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_{12}:C_4$ |