Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(289\)\(\medspace = 17^{2} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$b^{119}, b^{170}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{17}\times C_{238}$ |
| Order: | \(4046\)\(\medspace = 2 \cdot 7 \cdot 17^{2} \) |
| Exponent: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 17$ (hence hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_{17}^2$ |
| Order: | \(289\)\(\medspace = 17^{2} \) |
| Exponent: | \(17\) |
| Automorphism Group: | $\GL(2,17)$, of order \(78336\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 17 \) |
| Outer Automorphisms: | $\GL(2,17)$, of order \(78336\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 17 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_6\times C_{16}.\PSL(2,17).C_2$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(78336\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 17 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $17$ |
| Projective image | $C_{17}^2$ |