Subgroup ($H$) information
| Description: | $C_{35}$ |
| Order: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Index: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Generators: |
$b^{924}, b^{440}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{11}\times D_{140}$ |
| Order: | \(3080\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(1540\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $D_4\times C_{11}$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Automorphism Group: | $D_4\times C_{10}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{70}.C_{30}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2800\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{1540}$ | |||
| Normalizer: | $C_{11}\times D_{140}$ | |||
| Complements: | $D_4\times C_{11}$ | |||
| Minimal over-subgroups: | $C_{385}$ | $C_{70}$ | $D_{35}$ | $D_{35}$ |
| Maximal under-subgroups: | $C_7$ | $C_5$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{11}\times D_{140}$ |