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G = D81order 162 = 2·34

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D81, C81⋊C2, C27.S3, C3.D27, C9.1D9, sometimes denoted D162 or Dih81 or Dih162, SmallGroup(162,1)

Series: Derived Chief Lower central Upper central

C1C81 — D81
C1C3C9C27C81 — D81
C81 — D81
C1

Generators and relations for D81
 G = < a,b | a81=b2=1, bab=a-1 >

81C2
27S3
9D9
3D27

Smallest permutation representation of D81
On 81 points
Generators in S81
(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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81)
(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)

G:=sub<Sym(81)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81), (2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81), (2,81)(3,80)(4,79)(5,78)(6,77)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)], [(2,81),(3,80),(4,79),(5,78),(6,77),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42)])

D81 is a maximal subgroup of   D243  C81⋊C6  C81⋊S3
D81 is a maximal quotient of   Dic81  D243  C81⋊S3

42 conjugacy classes

class 1  2  3 9A9B9C27A···27I81A···81AA
order12399927···2781···81
size18122222···22···2

42 irreducible representations

dim112222
type++++++
imageC1C2S3D9D27D81
kernelD81C81C27C9C3C1
# reps1113927

Matrix representation of D81 in GL2(𝔽163) generated by

1653
11069
,
01
10
G:=sub<GL(2,GF(163))| [16,110,53,69],[0,1,1,0] >;

D81 in GAP, Magma, Sage, TeX

D_{81}
% in TeX

G:=Group("D81");
// GroupNames label

G:=SmallGroup(162,1);
// by ID

G=gap.SmallGroup(162,1);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,101,156,452,237,1803,138,2704]);
// Polycyclic

G:=Group<a,b|a^81=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D81 in TeX

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