Subgroup ($H$) information
| Description: | $C_{33}:C_4$ |
| Order: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| Generators: |
$ab, b^{8}, c^{3}, b^{12}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{132}.D_6$ |
| 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, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{12}$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{33}.(C_2^5\times C_{10})$ |
| $\operatorname{Aut}(H)$ | $D_6\times F_{11}$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_6\times F_{11}$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $S_3\times D_{11}$, of order \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||
| Normalizer: | $C_{132}.D_6$ | ||
| Minimal over-subgroups: | $C_{33}:C_{12}$ | $C_4\times D_{33}$ | |
| Maximal under-subgroups: | $C_{66}$ | $C_{11}:C_4$ | $C_3:C_4$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{33}:C_{12}$ |