Subgroup ($H$) information
| Description: | $C_{264}:C_2$ |
| Order: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Index: | \(3\) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Generators: |
$a^{3}, b^{24}, a^{2}, b^{66}, b^{33}, b^{132}$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a direct factor, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{264}:C_6$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{11}\times Q_8).C_{30}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^4\times F_{11}$, of order \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| $\operatorname{res}(S)$ | $C_2^4\times F_{11}$, of order \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $D_{22}$, of order \(44\)\(\medspace = 2^{2} \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_3\times D_{22}$ |