Subgroup ($H$) information
| Description: | $C_{264}$ |
| Order: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Generators: |
$c^{33}, c^{24}, c^{66}, c^{176}, c^{132}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), a semidirect factor, and cyclic (hence abelian, elementary ($p = 2,3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{24}:C_2\times F_{11}$ |
| Order: | \(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_{10}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $W$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Möbius function | $-2$ |
| Projective image | $D_{12}\times F_{11}$ |