Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(1216451004088320000\)\(\medspace = 2^{17} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(21,22)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2.A_{20}$ |
| Order: | \(2432902008176640000\)\(\medspace = 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| Exponent: | \(232792560\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $A_{20}$ |
| Order: | \(1216451004088320000\)\(\medspace = 2^{17} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| Exponent: | \(232792560\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| Automorphism Group: | Group of order \(2432902008176640000\)\(\medspace = 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $0$ |
The quotient is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 \(2432902008176640000\)\(\medspace = 2^{18} \cdot 3^{8} \cdot 5^{4} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \cdot 19 \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |