Subgroup ($H$) information
| Description: | $C_5:C_{24}$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$abc^{27}, bc^{30}, c^{60}, c^{80}, c^{24}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_{10}:C_{24}$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times C_2^6$, of order \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^3\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^3\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2\times C_{12}$ | ||
| Normalizer: | $C_{10}:C_{24}$ | ||
| Normal closure: | $C_{10}:C_{24}$ | ||
| Core: | $C_{60}$ | ||
| Minimal over-subgroups: | $C_{10}:C_{24}$ | ||
| Maximal under-subgroups: | $C_{60}$ | $C_5:C_8$ | $C_{24}$ |
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_{20}$ |