Subgroup ($H$) information
| Description: | $C_{20}.D_8$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Index: | $1$ |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$a, c^{10}, b^{6}c^{5}, c^{5}, b, c^{4}, b^{4}c^{10}$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, supersolvable (hence monomial), hyperelementary for $p = 2$, and metabelian.
Ambient group ($G$) information
| Description: | $C_{20}.D_8$ |
| 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2\times C_4\times C_2^2.C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_5:(C_2\times C_4\times C_2^2.C_2^5)$ |
| $W$ | $C_{10}.D_4$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |||
| Normalizer: | $C_{20}.D_8$ | |||
| Complements: | $C_1$ | |||
| Maximal under-subgroups: | $C_{20}:Q_8$ | $C_{20}:C_8$ | $C_{20}:C_8$ | $C_4.D_8$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{10}.D_4$ |