Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(18144\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(2,3)(4,6)(5,8)(7,9), (10,12,15,11,17,16,13)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\SL(2,8)^2$ |
| Order: | \(254016\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7^{2} \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, an A-group, and perfect (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | ${}^2G(2,3)\wr C_2$, of order \(4572288\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4536$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $16$ |
| Projective image | $\SL(2,8)^2$ |