Subgroup ($H$) information
| Description: | $C_{20}:C_4$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(1,5,4,3)(6,10,7,11)(8,13,9,12), (6,9,7,8)(10,13,11,12), (1,5,2,3,4)(6,9,7,8) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $A_5:Q_8$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_4\times S_5$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $C_{20}:C_4$ | |||
| Normal closure: | $A_5:Q_8$ | |||
| Core: | $C_4$ | |||
| Minimal over-subgroups: | $A_5:Q_8$ | |||
| Maximal under-subgroups: | $C_4\times D_5$ | $C_2\times F_5$ | $C_2\times F_5$ | $C_4:C_4$ |
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $-1$ |
| Projective image | $C_2\times S_5$ |