Subgroup ($H$) information
| Description: | $\PSL(2,169)$ |
| Order: | \(2413320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| Index: | $1$ |
| Exponent: | \(92820\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
73 & 133 \\
23 & 82
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
145 & 97 \\
2 & 142
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
60 & 10 \\
68 & 92
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is the commutator subgroup (hence characteristic and normal), the socle, a direct factor, nonabelian, a Hall subgroup, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $\PSL(2,169)$ |
| Order: | \(2413320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| Exponent: | \(92820\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,169).C_2^2$, 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 \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| $W$ | $\PSL(2,169)$, of order \(2413320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||||
| Normalizer: | $\PSL(2,169)$ | ||||||
| Complements: | $C_1$ | ||||||
| Maximal under-subgroups: | $C_{13}^2:C_{84}$ | $\PGL(2,13)$ | $\PGL(2,13)$ | $D_{85}$ | $D_{84}$ | $A_5$ | $A_5$ |
Other information
| Möbius function | $1$ |
| Projective image | $\PSL(2,169)$ |