Subgroup ($H$) information
| Description: | $C_5^4.C_4^2$ |
| Order: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Index: | \(2\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ab, b^{2}, d, a^{2}b^{8}c^{6}e^{2}, c^{4}e^{3}, e, c^{5}, c^{10}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{10}.D_5^2:F_5$ |
| Order: | \(20000\)\(\medspace = 2^{5} \cdot 5^{4} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.C_4^3.C_2.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_5^4.C_4^3.C_2^3.C_2^4$ |
| $W$ | $C_5^4:(C_2^2\times C_4)$, of order \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_5^4:(C_2^2\times C_4)$ |