Subgroup ($H$) information
| Description: | $C_{165}$ |
| Order: | \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \) |
| Index: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \) |
| Generators: |
$b^{1540}, b^{420}, b^{2772}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,5,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{11}\times D_{420}$ |
| Order: | \(9240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $D_{28}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $D_4\times F_7$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.C_4^2$, of order \(403200\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{20}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_{420}$ |