Subgroup ($H$) information
| Description: | $D_{154}$ |
| Order: | \(308\)\(\medspace = 2^{2} \cdot 7 \cdot 11 \) |
| Index: | \(17\) |
| Exponent: | \(154\)\(\medspace = 2 \cdot 7 \cdot 11 \) |
| Generators: |
$a, b^{1122}, b^{476}, b^{1309}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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: | $C_{17}$ |
| Order: | \(17\) |
| Exponent: | \(17\) |
| Automorphism Group: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| 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, and simple.
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 \(147840\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_{11}:(C_2\times C_{10}\times F_7)$ |
| $W$ | $D_{77}$, of order \(154\)\(\medspace = 2 \cdot 7 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{34}$ | ||||
| Normalizer: | $C_{17}\times D_{154}$ | ||||
| Complements: | $C_{17}$ | ||||
| Minimal over-subgroups: | $C_{17}\times D_{154}$ | ||||
| Maximal under-subgroups: | $C_{154}$ | $D_{77}$ | $D_{77}$ | $D_{22}$ | $D_{14}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{17}\times D_{77}$ |