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G = D89order 178 = 2·89

Dihedral group

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

Aliases: D89, C89⋊C2, sometimes denoted D178 or Dih89 or Dih178, SmallGroup(178,1)

Series: Derived Chief Lower central Upper central

C1C89 — D89
C1C89 — D89
C89 — D89
C1

Generators and relations for D89
 G = < a,b | a89=b2=1, bab=a-1 >

89C2

Smallest permutation representation of D89
On 89 points: primitive
Generators in S89
(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 82 83 84 85 86 87 88 89)
(1 89)(2 88)(3 87)(4 86)(5 85)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)

G:=sub<Sym(89)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89)], [(1,89),(2,88),(3,87),(4,86),(5,85),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,73),(18,72),(19,71),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46)])

D89 is a maximal subgroup of   C89⋊C4
D89 is a maximal quotient of   Dic89

46 conjugacy classes

class 1  2 89A···89AR
order1289···89
size1892···2

46 irreducible representations

dim112
type+++
imageC1C2D89
kernelD89C89C1
# reps1144

Matrix representation of D89 in GL2(𝔽179) generated by

147178
10
,
147178
12832
G:=sub<GL(2,GF(179))| [147,1,178,0],[147,128,178,32] >;

D89 in GAP, Magma, Sage, TeX

D_{89}
% in TeX

G:=Group("D89");
// GroupNames label

G:=SmallGroup(178,1);
// by ID

G=gap.SmallGroup(178,1);
# by ID

G:=PCGroup([2,-2,-89,705]);
// Polycyclic

G:=Group<a,b|a^89=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D89 in TeX

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