Subgroup ($H$) information
| Description: | $Q_8:D_5$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$a, c^{2}, c^{4}, c, d^{2}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{20}.C_2^4$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{10}.(C_2^4\times S_4)$, of order \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_{10}.C_2^4$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{10}.C_2^4$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_5:D_4$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
| Centralizer: | $Q_8$ | |||
| Normalizer: | $C_{20}.C_2^4$ | |||
| Complements: | $C_2^2$ | |||
| Minimal over-subgroups: | $C_{20}.D_4$ | |||
| Maximal under-subgroups: | $D_{20}$ | $C_5\times Q_8$ | $C_5:C_8$ | $\SD_{16}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $C_2^3:D_{10}$ |