Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(3\) |
| Generators: |
$c^{44}$
|
| 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 $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{66}:C_{10}^2$ |
| Order: | \(6600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{22}:C_{10}^2$ |
| Order: | \(2200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $C_2^2.C_{165}.C_{10}.C_2^3$ |
| Outer Automorphisms: | $F_5\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_{165}.C_{60}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{110}:C_{30}$ | ||||
| Normalizer: | $C_{66}:C_{10}^2$ | ||||
| Complements: | $C_{22}:C_{10}^2$ | ||||
| Minimal over-subgroups: | $C_{33}$ | $C_{15}$ | $C_{15}$ | $C_6$ | $S_3$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $440$ |
| Projective image | $C_{66}:C_{10}^2$ |