Subgroup ($H$) information
| Description: | $C_{66}:C_{12}$ |
| Order: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Generators: |
$ac^{33}, c^{12}, b^{2}, b^{3}c^{66}, c^{66}, c^{88}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $D_{66}:C_{12}$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_6\times C_{11}:C_5).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_{66}.C_{10}.C_2^4$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{66}$, of order \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{66}$ |