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G = D157order 314 = 2·157

Dihedral group

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

Aliases: D157, C157⋊C2, sometimes denoted D314 or Dih157 or Dih314, SmallGroup(314,1)

Series: Derived Chief Lower central Upper central

C1C157 — D157
C1C157 — D157
C157 — D157
C1

Generators and relations for D157
 G = < a,b | a157=b2=1, bab=a-1 >

157C2

Smallest permutation representation of D157
On 157 points: primitive
Generators in S157
(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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157)
(1 157)(2 156)(3 155)(4 154)(5 153)(6 152)(7 151)(8 150)(9 149)(10 148)(11 147)(12 146)(13 145)(14 144)(15 143)(16 142)(17 141)(18 140)(19 139)(20 138)(21 137)(22 136)(23 135)(24 134)(25 133)(26 132)(27 131)(28 130)(29 129)(30 128)(31 127)(32 126)(33 125)(34 124)(35 123)(36 122)(37 121)(38 120)(39 119)(40 118)(41 117)(42 116)(43 115)(44 114)(45 113)(46 112)(47 111)(48 110)(49 109)(50 108)(51 107)(52 106)(53 105)(54 104)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)

G:=sub<Sym(157)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157), (1,157)(2,156)(3,155)(4,154)(5,153)(6,152)(7,151)(8,150)(9,149)(10,148)(11,147)(12,146)(13,145)(14,144)(15,143)(16,142)(17,141)(18,140)(19,139)(20,138)(21,137)(22,136)(23,135)(24,134)(25,133)(26,132)(27,131)(28,130)(29,129)(30,128)(31,127)(32,126)(33,125)(34,124)(35,123)(36,122)(37,121)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,113)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(57,101)(58,100)(59,99)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157), (1,157)(2,156)(3,155)(4,154)(5,153)(6,152)(7,151)(8,150)(9,149)(10,148)(11,147)(12,146)(13,145)(14,144)(15,143)(16,142)(17,141)(18,140)(19,139)(20,138)(21,137)(22,136)(23,135)(24,134)(25,133)(26,132)(27,131)(28,130)(29,129)(30,128)(31,127)(32,126)(33,125)(34,124)(35,123)(36,122)(37,121)(38,120)(39,119)(40,118)(41,117)(42,116)(43,115)(44,114)(45,113)(46,112)(47,111)(48,110)(49,109)(50,108)(51,107)(52,106)(53,105)(54,104)(55,103)(56,102)(57,101)(58,100)(59,99)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157)], [(1,157),(2,156),(3,155),(4,154),(5,153),(6,152),(7,151),(8,150),(9,149),(10,148),(11,147),(12,146),(13,145),(14,144),(15,143),(16,142),(17,141),(18,140),(19,139),(20,138),(21,137),(22,136),(23,135),(24,134),(25,133),(26,132),(27,131),(28,130),(29,129),(30,128),(31,127),(32,126),(33,125),(34,124),(35,123),(36,122),(37,121),(38,120),(39,119),(40,118),(41,117),(42,116),(43,115),(44,114),(45,113),(46,112),(47,111),(48,110),(49,109),(50,108),(51,107),(52,106),(53,105),(54,104),(55,103),(56,102),(57,101),(58,100),(59,99),(60,98),(61,97),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80)])

80 conjugacy classes

class 1  2 157A···157BZ
order12157···157
size11572···2

80 irreducible representations

dim112
type+++
imageC1C2D157
kernelD157C157C1
# reps1178

Matrix representation of D157 in GL2(𝔽1571) generated by

6661570
11251140
,
2681332
15561303
G:=sub<GL(2,GF(1571))| [666,1125,1570,1140],[268,1556,1332,1303] >;

D157 in GAP, Magma, Sage, TeX

D_{157}
% in TeX

G:=Group("D157");
// GroupNames label

G:=SmallGroup(314,1);
// by ID

G=gap.SmallGroup(314,1);
# by ID

G:=PCGroup([2,-2,-157,1249]);
// Polycyclic

G:=Group<a,b|a^157=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D157 in TeX

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