Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(9792\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 17 \) |
| Exponent: | \(2\) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{61} & \alpha^{107} \\ \alpha^{64} & \alpha^{205} \\ \end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_4.\PGL(2,17)$ |
| Order: | \(19584\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 17 \) |
| Exponent: | \(4896\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \PSL(2,17).C_2$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_{36}$ | ||
| Normalizer: | $C_2\times C_{36}$ | ||
| Normal closure: | $\SL(2,17):C_2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_6$ | $S_3$ | $C_2^2$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this conjugacy class | $272$ |
| Möbius function | $0$ |
| Projective image | $C_4.\PGL(2,17)$ |