Subgroup ($H$) information
| Description: | $D_{35}$ |
| Order: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Index: | \(19\) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$a, b^{285}, b^{266}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{19}\times D_{35}$ |
| Order: | \(1330\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 19 \) |
| Exponent: | \(1330\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_{19}$ |
| Order: | \(19\) |
| Exponent: | \(19\) |
| Automorphism Group: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Outer Automorphisms: | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{18}\times F_5\times F_7$ |
| $\operatorname{Aut}(H)$ | $F_5\times F_7$, of order \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_5\times F_7$, of order \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $D_{35}$, of order \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{19}$ | ||
| Normalizer: | $C_{19}\times D_{35}$ | ||
| Complements: | $C_{19}$ | ||
| Minimal over-subgroups: | $C_{19}\times D_{35}$ | ||
| Maximal under-subgroups: | $C_{35}$ | $D_7$ | $D_5$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{19}\times D_{35}$ |