Subgroup ($H$) information
| Description: | $C_{25}$ |
| Order: | \(25\)\(\medspace = 5^{2} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(25\)\(\medspace = 5^{2} \) |
| Generators: |
$d^{88}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $5$-Sylow subgroup (hence a Hall subgroup), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_2^4.C_{100}$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2:\OD_{16}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^7:D_4$, of order \(1024\)\(\medspace = 2^{10} \) |
| Outer Automorphisms: | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^4.C_2^5.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $C_{20}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{20}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1024\)\(\medspace = 2^{10} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2^4.C_{100}$ | |||
| Normalizer: | $C_2^4.C_{100}$ | |||
| Complements: | $C_2^2:\OD_{16}$ | |||
| Minimal over-subgroups: | $C_{50}$ | $C_{50}$ | $C_{50}$ | $C_{50}$ |
| Maximal under-subgroups: | $C_5$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^2:\OD_{16}$ |