Subgroup ($H$) information
| Description: | $D_{24}$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$abc, c^{16}, b^{2}c^{18}, c^{12}, c^{24}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_2^3.D_{24}$ |
| Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:((C_2\times C_4^2).C_2^6)$ |
| $\operatorname{Aut}(H)$ | $D_8:D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_8:D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_{24}$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||
| Normalizer: | $D_{24}:C_4$ | ||
| Normal closure: | $C_2\times D_{24}$ | ||
| Core: | $C_{24}$ | ||
| Minimal over-subgroups: | $C_2\times D_{24}$ | ||
| Maximal under-subgroups: | $C_{24}$ | $D_{12}$ | $D_8$ |
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_2^2.D_{24}$ |