Subgroup ($H$) information
| Description: | $C_{13}:F_{13}$ |
| Order: | \(2028\)\(\medspace = 2^{2} \cdot 3 \cdot 13^{2} \) |
| Index: | \(2380\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \cdot 17 \) |
| Exponent: | \(156\)\(\medspace = 2^{2} \cdot 3 \cdot 13 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
124 & 161 \\
139 & 51
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
136 & 13 \\
110 & 67
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
125 & 22 \\
119 & 127
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
120 & 147 \\
76 & 62
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
140 & 43 \\
21 & 8
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PSL(2,169).C_2^2$, of order \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $C_{13}^2.\GL(2,13)$, of order \(4429152\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 13^{3} \) |
| $W$ | $F_{169}$, of order \(28392\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $F_{169}$ | ||
| Normal closure: | $\PSL(2,169)$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_{13}^2:C_{84}$ | $C_{13}^2:C_{24}$ | |
| Maximal under-subgroups: | $C_{13}^2:C_6$ | $C_{13}^2:C_4$ | $F_{13}$ |
Other information
| Number of subgroups in this conjugacy class | $170$ |
| Möbius function | $0$ |
| Projective image | $\PGL(2,169)$ |