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G = D66  order 132 = 22·3·11

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D66, C2×D33, C22⋊S3, C6⋊D11, C3⋊2D22, C11⋊2D6, C66⋊1C2, C33⋊2C22, sometimes denoted D132 or Dih66 or Dih132, SmallGroup(132,9)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — D66
Chief series
C1 — C11 — C33 — D33 — D66
Lower central
C33 — D66
Upper central
C1 — C2

Generators and relations for D66
 G = < a,b | a66=b2=1, bab=a-1 >

C1
C2
33C2
33C2
C3
C11
33C22
C6
11S3
11S3
C22
3D11
3D11
C33
11D6
3D22
D33
D33
C66
D66

Smallest permutation representation of D66
►On 66 points
Generators in S66
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66)

(1 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)
 
G:=sub<Sym(66)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)], [(1,66),(2,65),(3,64),(4,63),(5,62),(6,61),(7,60),(8,59),(9,58),(10,57),(11,56),(12,55),(13,54),(14,53),(15,52),(16,51),(17,50),(18,49),(19,48),(20,47),(21,46),(22,45),(23,44),(24,43),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34)]])
 

D66 is a maximal subgroup of   D33⋊C4  C3⋊D44  C11⋊D12  D132  C33⋊7D4  C2×S3×D11
D66 is a maximal quotient of   Dic66  D132  C33⋊7D4

36 conjugacy classes

class 1 2A2B2C 3  6 11A···11E22A···22E33A···33J66A···66J
order12223611···1122···2233···3366···66
size113333222···22···22···22···2

36 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D11D22D33D66
kernelD66D33C66C22C11C6C3C2C1
# reps12111551010

Matrix representation of D66 ►in GL2(𝔽67) generated by

4415
3515
,
5414
5513
G:=sub<GL(2,GF(67))| [44,35,15,15],[54,55,14,13] >;
 

D66 in GAP, Magma, Sage, TeX 

D_{66}
 
% in TeX
 
G:=Group("D66");
 
// GroupNames label
 
G:=SmallGroup(132,9);
 
// by ID
 
G=gap.SmallGroup(132,9);
 
# by ID
 
G:=PCGroup([4,-2,-2,-3,-11,98,1923]);
 
// Polycyclic
 
G:=Group<a,b|a^66=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D66 in TeX

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