Subgroup ($H$) information
| Description: | $C_{119}$ |
| Order: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Index: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Generators: |
$b^{1122}, b^{154}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
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: | $D_{22}$ |
| Order: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| 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 \(147840\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{2618}$ | |||
| Normalizer: | $C_{17}\times D_{154}$ | |||
| Complements: | $D_{22}$ | |||
| Minimal over-subgroups: | $C_{1309}$ | $C_{238}$ | $D_7\times C_{17}$ | $D_7\times C_{17}$ |
| Maximal under-subgroups: | $C_{17}$ | $C_7$ |
Other information
| Möbius function | $-22$ |
| Projective image | $D_{154}$ |