Subgroup ($H$) information
| Description: | $C_5^2$ |
| Order: | \(25\)\(\medspace = 5^{2} \) |
| Index: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(5\) |
| Generators: |
$b^{4}, de^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_5^6:\OD_{16}$ |
| Order: | \(250000\)\(\medspace = 2^{4} \cdot 5^{6} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_5^4.\OD_{16}$ |
| Order: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Automorphism Group: | $C_5^4:(C_4^3:C_2^2)$, of order \(160000\)\(\medspace = 2^{8} \cdot 5^{4} \) |
| Outer Automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(32000000\)\(\medspace = 2^{11} \cdot 5^{6} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_5\times C_5^5:C_2^2$ |
| Normalizer: | $C_5^6:\OD_{16}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5^6:\OD_{16}$ |