Subgroup ($H$) information
| Description: | $C_{132}$ |
| Order: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Index: | \(2\) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Generators: |
$ab^{33}, b^{12}, b^{66}, b^{88}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, maximal, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $Q_8\times C_{33}$ |
| Order: | \(264\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence 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 $1$ |
| Nilpotency class: | $1$ |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{10}\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{10}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^2\times C_{10}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{132}$ | ||
| Normalizer: | $Q_8\times C_{33}$ | ||
| Minimal over-subgroups: | $Q_8\times C_{33}$ | ||
| Maximal under-subgroups: | $C_{66}$ | $C_{44}$ | $C_{12}$ |
| Autjugate subgroups: | 264.30.2.a1.a1 | 264.30.2.a1.b1 |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2^2$ |