Subgroup ($H$) information
| Description: | $\PSL(2,169)$ |
| Order: | \(2413320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| Index: | \(2\) |
| Exponent: | \(92820\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
8 & -1 \\
154 & 8
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
62 & -1 \\
101 & 84
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
164 & -1 \\
141 & 82
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
112 & 67 \\
149 & 120
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
5 & -1 \\
112 & 5
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, maximal, a semidirect factor, nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $\PGL(2,169)$ |
| Order: | \(4826640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| Exponent: | \(185640\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,169).C_2^2$, of order \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $\PSL(2,169).C_2^2$, of order \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| $W$ | $\PGL(2,169)$, of order \(4826640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $\PGL(2,169)$ | ||||
| Complements: | $C_2$ | ||||
| Minimal over-subgroups: | $\PGL(2,169)$ | ||||
| Maximal under-subgroups: | $C_{13}^2:C_{84}$ | $\PGL(2,13)$ | $D_{85}$ | $D_{84}$ | $A_5$ |
Other information
| Möbius function | $-1$ |
| Projective image | $\PGL(2,169)$ |