Subgroup ($H$) information
| Description: | $C_3:D_8$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\left(\begin{array}{rr}
11 & 45 \\
72 & 17
\end{array}\right), \left(\begin{array}{rr}
1 & 28 \\
0 & 1
\end{array}\right), \left(\begin{array}{rr}
1 & 12 \\
0 & 41
\end{array}\right), \left(\begin{array}{rr}
13 & 72 \\
48 & 1
\end{array}\right), \left(\begin{array}{rr}
13 & 0 \\
0 & 13
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $D_{12}:S_4$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $D_4\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4\times D_6$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_3:D_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_6:D_8$ | |||
| Normal closure: | $D_{12}:S_4$ | |||
| Core: | $D_{12}$ | |||
| Minimal over-subgroups: | $D_{12}:S_3$ | $C_6:D_8$ | ||
| Maximal under-subgroups: | $D_{12}$ | $C_3\times D_4$ | $C_3:C_8$ | $D_8$ |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $2$ |
| Projective image | $D_6:S_4$ |