Subgroup ($H$) information
| Description: | $C_{10}\times D_{66}$ |
| Order: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(5\) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, c^{6}, c^{33}, a^{2}, b^{5}, c^{44}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
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: | $C_5$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_{165}.C_{60}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_{33}\times A_4).C_{20}.C_2^3$ |
| $W$ | $C_{33}:C_{10}$, of order \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_{33}:C_{10}$ |