Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2097152\)\(\medspace = 2^{21} \) |
| Index: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | not computed |
| Generators: |
$\langle(37,38)(39,40), (33,34)(35,36)(41,42)(43,44), (5,8,6,7)(13,16,14,15)(17,18) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.C_2^{10}.M_{11}$ |
| Order: | \(33218887680\)\(\medspace = 2^{26} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2\times M_{11}$ |
| Order: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $1$ |
The quotient is nonabelian and nonsolvable.
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 \(66437775360\)\(\medspace = 2^{27} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | 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 |