Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(748\)\(\medspace = 2^{2} \cdot 11 \cdot 17 \) |
| Exponent: | \(7\) |
| Generators: |
$b^{1122}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{17}\times D_{154}$ |
| Order: | \(5236\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \cdot 17 \) |
| Exponent: | \(2618\)\(\medspace = 2 \cdot 7 \cdot 11 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{17}\times D_{22}$ |
| Order: | \(748\)\(\medspace = 2^{2} \cdot 11 \cdot 17 \) |
| Exponent: | \(374\)\(\medspace = 2 \cdot 11 \cdot 17 \) |
| Automorphism Group: | $C_{11}.C_{80}.C_2^2$ |
| Outer Automorphisms: | $C_2\times C_{80}$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:C_2^2$, of order \(147840\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{2618}$ | ||||
| Normalizer: | $C_{17}\times D_{154}$ | ||||
| Complements: | $C_{17}\times D_{22}$ | ||||
| Minimal over-subgroups: | $C_{119}$ | $C_{77}$ | $C_{14}$ | $D_7$ | $D_7$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $22$ |
| Projective image | $C_{17}\times D_{154}$ |