Subgroup ($H$) information
| Description: | $A_4:Q_8$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(5\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a, d^{5}, cd^{5}, b^{3}, b^{8}, b^{6}$
|
| Derived length: | $3$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, and a $96$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial).
Ambient group ($G$) information
| Description: | $C_{20}.S_4$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times D_4\times S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{10}$ | ||||
| Normalizer: | $C_{20}.S_4$ | ||||
| Complements: | $C_5$ | ||||
| Minimal over-subgroups: | $C_{20}.S_4$ | ||||
| Maximal under-subgroups: | $C_4\times A_4$ | $A_4:C_4$ | $A_4:C_4$ | $C_2^2:Q_8$ | $C_3:Q_8$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{10}\times S_4$ |