Subgroup ($H$) information
| Description: | $C_{43}$ |
| Order: | \(43\) |
| Index: | \(4300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 43 \) |
| Exponent: | \(43\) |
| Generators: |
$b^{5}$
|
| 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 $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{215}:C_{860}$ |
| Order: | \(184900\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 43^{2} \) |
| Exponent: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_5:C_{860}$ |
| Order: | \(4300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 43 \) |
| Exponent: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Automorphism Group: | $C_2^2\times C_{84}\times F_5$ |
| Outer Automorphisms: | $C_2^3\times C_{84}$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{215}.C_7^2.C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{42}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{215}\times C_{430}$ | ||||
| Normalizer: | $C_{215}:C_{860}$ | ||||
| Complements: | $C_5:C_{860}$ | ||||
| Minimal over-subgroups: | $C_{43}^2$ | $C_{215}$ | $C_{215}$ | $C_{215}$ | $C_{86}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{215}:C_{860}$ |