Subgroup ($H$) information
| Description: | $\PSL(2,1849)$ |
| Order: | \(3160680600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \) |
| Index: | $1$ |
| Exponent: | \(36752100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
1 & -1 \\
-1 & 1847
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
924 & 0 \\
924 & -1
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, and simple (hence nonsolvable, perfect, quasisimple, and almost simple). Whether it is a direct factor has not been computed.
Ambient group ($G$) information
| Description: | $\PSL(2,1849)$ |
| Order: | \(3160680600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \) |
| Exponent: | \(36752100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(12642722400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \) |
| $\operatorname{Aut}(H)$ | Group of order \(12642722400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 37 \cdot 43^{2} \) |
| $\card{W}$ | 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 |