Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2500\)\(\medspace = 2^{2} \cdot 5^{4} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | not computed |
| Generators: |
$abc^{4}de^{2}, d^{2}, c^{2}, e, c^{5}, b^{2}e$
|
| Derived length: | not computed |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $D_5^4$ |
| Order: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:C_4\wr S_4$, of order \(3840000\)\(\medspace = 2^{11} \cdot 3 \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $D_{10}^2$, of order \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $2$ |
| Projective image | $D_5^4$ |