Subgroup ($H$) information
| Description: | $C_{66}:C_{10}$ |
| Order: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(5\) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, a^{2}, b^{110}, b^{165}, b^{30}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{330}:C_{10}$ |
| Order: | \(3300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{55}.C_{30}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_6\times F_{11}$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $\operatorname{res}(S)$ | $D_6\times F_{11}$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_{33}:C_{10}$, of order \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $5$ |
| Number of conjugacy classes in this autjugacy class | $5$ |
| Möbius function | $-1$ |
| Projective image | $C_{165}:C_{10}$ |