Subgroup ($H$) information
| Description: | $C_4.C_{10}^2$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a, c^{10}, c^{5}, b^{4}, b^{10}c^{10}, c^{4}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), maximal, a semidirect factor, nonabelian, and metabelian.
Ambient group ($G$) information
| Description: | $C_{10}^2.C_2^3$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| 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 $1$ |
| Nilpotency class: | $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^2\times C_4^2\times C_2^2\wr C_2)$ |
| $\operatorname{Aut}(H)$ | $C_2^2:S_4:C_2\times \GL(2,5)$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^4.C_2^5$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{10}$ |