Subgroup ($H$) information
| Description: | $C_2\times C_{28}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{10} & \alpha^{63} \\ \alpha^{167} & \alpha^{94} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{132} & 0 \\ 0 & \alpha^{132} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{150} & 0 \\ 0 & \alpha^{150} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{24} & 0 \\ 0 & \alpha^{24} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $\SL(2,13):C_{14}$ |
| Order: | \(30576\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{2} \cdot 13 \) |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $1$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6\times \PSL(2,13).C_2$, of order \(26208\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_6\times D_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $91$ |
| Möbius function | $6$ |
| Projective image | $\PSL(2,13)$ |