Subgroup ($H$) information
| Description: | $C_2.\PGL(2,17)$ |
| Order: | \(9792\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 17 \) |
| Index: | \(2\) |
| Exponent: | \(4896\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17 \) |
| Generators: |
$\left(\begin{array}{ll}\alpha^{78} & \alpha^{256} \\ \alpha^{159} & \alpha^{50} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{259} & \alpha^{202} \\ \alpha^{67} & \alpha^{115} \\ \end{array}\right), \left(\begin{array}{ll}\alpha^{144} & 0 \\ 0 & \alpha^{144} \\ \end{array}\right)$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and nonsolvable.
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.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient 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.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times \PSL(2,17).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times \PGL(2,17)$, of order \(9792\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $W$ | $\PGL(2,17)$, of order \(4896\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 17 \) |
Related subgroups
| Centralizer: | $C_4$ | |||
| Normalizer: | $C_4.\PGL(2,17)$ | |||
| Complements: | $C_2$ $C_2$ | |||
| Minimal over-subgroups: | $C_4.\PGL(2,17)$ | |||
| Maximal under-subgroups: | $\SL(2,17)$ | $C_{17}:C_{32}$ | $C_9:Q_8$ | $Q_{64}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2\times \PGL(2,17)$ |