Subgroup ($H$) information
| Description: | $C_5:D_5$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Index: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{2}b^{2}d^{4}e^{3}f^{2}, c, b^{8}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_5^5.C_{40}:C_4$ |
| Order: | \(500000\)\(\medspace = 2^{5} \cdot 5^{6} \) |
| 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 \(16\)\(\medspace = 2^{4} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^6.C_4^2.C_2^3.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_5^2.\GL(2,5)$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
| $W$ | $F_5^2$, of order \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_5^4:C_2$ |
| Normalizer: | $C_5^5.C_{40}:C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5^5.C_{40}:C_4$ |