Subgroup ($H$) information
| Description: | $C_{132}.C_{10}$ |
| Order: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}bc^{133}, c^{24}, a^{2}, c^{132}, c^{66}, c^{176}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{66}.C_{10}.C_2^3$ |
| $W$ | $D_{12}\times F_{11}$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $2$ |
| Projective image | $D_{12}\times F_{11}$ |