Subgroup ($H$) information
| Description: | $C_2^{12}$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Index: | \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(17,18)(19,20)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44), (37,38)(39,40) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the radical, abelian (hence metabelian and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor, a semidirect factor, or almost simple has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.M_{11}$ |
| Order: | \(32440320\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $M_{11}$ |
| Order: | \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| 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: | $0$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{12}.M_{11}$, of order \(32440320\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | Group of order \(644\!\cdots\!000\)\(\medspace = 2^{66} \cdot 3^{8} \cdot 5^{3} \cdot 7^{4} \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 31^{2} \cdot 73 \cdot 89 \cdot 127 \) |
| $W$ | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_2^{12}$ |
| Normalizer: | $C_2^{12}.M_{11}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^{11}.M_{11}$ |