Subgroup ($H$) information
| Description: | $C_{13}^2:C_{84}$ |
| Order: | \(14196\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13^{2} \) |
| Index: | \(340\)\(\medspace = 2^{2} \cdot 5 \cdot 17 \) |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Generators: |
$\left[ \left(\begin{array}{rr}
124 & 161 \\
139 & 51
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
125 & 1 \\
98 & 150
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
51 & 144 \\
73 & 0
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
167 & 81 \\
10 & 27
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
143 & 131 \\
60 & 15
\end{array}\right) \right], \left[ \left(\begin{array}{rr}
140 & 43 \\
21 & 8
\end{array}\right) \right]$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), 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 \(9653280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $C_{13}^2.C_{168}.C_2$ |
| $W$ | $F_{169}$, of order \(28392\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $170$ |
| Möbius function | $1$ |
| Projective image | $\PGL(2,169)$ |