Copied to
clipboard

G = D87order 174 = 2·3·29

Dihedral group

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

Aliases: D87, C29⋊S3, C3⋊D29, C871C2, sometimes denoted D174 or Dih87 or Dih174, SmallGroup(174,3)

Series: Derived Chief Lower central Upper central

C1C87 — D87
C1C29C87 — D87
C87 — D87
C1

Generators and relations for D87
 G = < a,b | a87=b2=1, bab=a-1 >

87C2
29S3
3D29

Smallest permutation representation of D87
On 87 points
Generators in S87
(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)
(1 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)

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

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

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

D87 is a maximal subgroup of   S3×D29
D87 is a maximal quotient of   Dic87

45 conjugacy classes

class 1  2  3 29A···29N87A···87AB
order12329···2987···87
size18722···22···2

45 irreducible representations

dim11222
type+++++
imageC1C2S3D29D87
kernelD87C87C29C3C1
# reps1111428

Matrix representation of D87 in GL2(𝔽349) generated by

46103
104286
,
72100
182277
G:=sub<GL(2,GF(349))| [46,104,103,286],[72,182,100,277] >;

D87 in GAP, Magma, Sage, TeX

D_{87}
% in TeX

G:=Group("D87");
// GroupNames label

G:=SmallGroup(174,3);
// by ID

G=gap.SmallGroup(174,3);
# by ID

G:=PCGroup([3,-2,-3,-29,25,1514]);
// Polycyclic

G:=Group<a,b|a^87=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D87 in TeX

׿
×
𝔽