Subgroup ($H$) information
| Description: | $C_{215}$ |
| Order: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Index: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Exponent: | \(215\)\(\medspace = 5 \cdot 43 \) |
| Generators: |
$b^{86}, b^{5}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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: | $C_{860}$ |
| Order: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Exponent: | \(860\)\(\medspace = 2^{2} \cdot 5 \cdot 43 \) |
| Automorphism Group: | $C_2^2\times C_{84}$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2^2\times C_{84}$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,43$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{215}.C_7^2.C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{84}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{215}\times C_{430}$ | ||
| Normalizer: | $C_{215}:C_{860}$ | ||
| Complements: | $C_{860}$ | ||
| Minimal over-subgroups: | $C_{43}\times C_{215}$ | $C_5\times C_{215}$ | $C_{430}$ |
| Maximal under-subgroups: | $C_{43}$ | $C_5$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{215}:C_{860}$ |