Subgroup ($H$) information
| Description: | $C_{20}:C_4$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(5\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a^{5}, a^{10}, b^{10}, b^{5}, b^{4}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{20}:C_{20}$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{20}:C_4^2$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_{10}$ | |||
| Normalizer: | $C_{20}:C_{20}$ | |||
| Complements: | $C_5$ $C_5$ | |||
| Minimal over-subgroups: | $C_{20}:C_{20}$ | |||
| Maximal under-subgroups: | $C_4\times D_5$ | $C_2\times F_5$ | $C_2\times F_5$ | $C_4:C_4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{10}\times F_5$ |