Subgroup ($H$) information
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$a^{2}, b^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{66}:C_4$ |
| Order: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{10}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_{66}$ | |||
| Normalizer: | $C_{66}:C_4$ | |||
| Minimal over-subgroups: | $C_{66}$ | $C_2\times C_{22}$ | $C_{11}:C_4$ | $C_{11}:C_4$ |
| Maximal under-subgroups: | $C_{11}$ | $C_2$ |
Other information
| Möbius function | $-6$ |
| Projective image | $D_{66}$ |