Subgroup ($H$) information
| Description: | $C_3\times D_{66}$ |
| Order: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$a^{3}, a^{2}, b^{176}, b^{132}, b^{24}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| 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) and a $p$-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{33}.(C_2^5\times C_{10})$ |
| $\operatorname{Aut}(H)$ | $C_{11}:(C_2^2\times C_{10}\times S_3)$ |
| $\card{\operatorname{res}(\operatorname{Aut}(G))}$ | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $D_{66}$, of order \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_{12}$ | ||||
| Normalizer: | $C_{264}:C_6$ | ||||
| Minimal over-subgroups: | $C_{12}\times D_{33}$ | ||||
| Maximal under-subgroups: | $C_3\times C_{66}$ | $C_3\times D_{33}$ | $D_{66}$ | $C_3\times D_{22}$ | $C_6\times S_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_4\times D_{33}$ |