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G = D86order 172 = 22·43

Dihedral group

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

Aliases: D86, C2×D43, C86⋊C2, C43⋊C22, sometimes denoted D172 or Dih86 or Dih172, SmallGroup(172,3)

Series: Derived Chief Lower central Upper central

C1C43 — D86
C1C43D43 — D86
C43 — D86
C1C2

Generators and relations for D86
 G = < a,b | a86=b2=1, bab=a-1 >

43C2
43C2
43C22

Smallest permutation representation of D86
On 86 points
Generators in S86
(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)
(1 86)(2 85)(3 84)(4 83)(5 82)(6 81)(7 80)(8 79)(9 78)(10 77)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)

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

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

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

D86 is a maximal subgroup of   D172  C43⋊D4
D86 is a maximal quotient of   Dic86  D172  C43⋊D4

46 conjugacy classes

class 1 2A2B2C43A···43U86A···86U
order122243···4386···86
size1143432···22···2

46 irreducible representations

dim11122
type+++++
imageC1C2C2D43D86
kernelD86D43C86C2C1
# reps1212121

Matrix representation of D86 in GL2(𝔽173) generated by

168113
5129
,
154148
4919
G:=sub<GL(2,GF(173))| [168,5,113,129],[154,49,148,19] >;

D86 in GAP, Magma, Sage, TeX

D_{86}
% in TeX

G:=Group("D86");
// GroupNames label

G:=SmallGroup(172,3);
// by ID

G=gap.SmallGroup(172,3);
# by ID

G:=PCGroup([3,-2,-2,-43,1514]);
// Polycyclic

G:=Group<a,b|a^86=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D86 in TeX

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