Subgroup ($H$) information
| Description: | $C_4.C_4^2$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(25\)\(\medspace = 5^{2} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b^{5}, c^{25}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_4.C_{20}^2$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_5^2$ |
| Order: | \(25\)\(\medspace = 5^{2} \) |
| Exponent: | \(5\) |
| Automorphism Group: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.C_2^6.C_2^2.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2^5.D_4^2$, of order \(2048\)\(\medspace = 2^{11} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^5.D_4^2$, of order \(2048\)\(\medspace = 2^{11} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2\times C_{10}\times C_{20}$ | ||
| Normalizer: | $C_4.C_{20}^2$ | ||
| Complements: | $C_5^2$ | ||
| Minimal over-subgroups: | $C_4^2.C_{20}$ | ||
| Maximal under-subgroups: | $C_2\times C_4^2$ | $C_2^2\times C_8$ | $C_8:C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $5$ |
| Projective image | $C_{10}^2$ |