Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(3\) |
| Generators: |
$c^{22}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is 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: | $(C_3\times C_{33}):C_{10}$ |
| Order: | \(990\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| 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: | $C_{33}:C_{10}$ |
| Order: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $S_3\times F_{11}$, of order \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{11}\times C_3^2:\GL(2,3)$, of order \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5940\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $33$ |
| Projective image | $(C_3\times C_{33}):C_{10}$ |