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G = D156order 312 = 23·3·13

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D156, C4⋊D39, C31D52, C394D4, C521S3, C131D12, C1561C2, D781C2, C121D13, C2.4D78, C6.10D26, C26.10D6, C78.10C22, sometimes denoted D312 or Dih156 or Dih312, SmallGroup(312,39)

Series: Derived Chief Lower central Upper central

C1C78 — D156
C1C13C39C78D78 — D156
C39C78 — D156
C1C2C4

Generators and relations for D156
 G = < a,b | a156=b2=1, bab=a-1 >

78C2
78C2
39C22
39C22
26S3
26S3
6D13
6D13
39D4
13D6
13D6
3D26
3D26
2D39
2D39
13D12
3D52

Smallest permutation representation of D156
On 156 points
Generators in S156
(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)
(1 156)(2 155)(3 154)(4 153)(5 152)(6 151)(7 150)(8 149)(9 148)(10 147)(11 146)(12 145)(13 144)(14 143)(15 142)(16 141)(17 140)(18 139)(19 138)(20 137)(21 136)(22 135)(23 134)(24 133)(25 132)(26 131)(27 130)(28 129)(29 128)(30 127)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(37 120)(38 119)(39 118)(40 117)(41 116)(42 115)(43 114)(44 113)(45 112)(46 111)(47 110)(48 109)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)

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

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

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

81 conjugacy classes

class 1 2A2B2C 3  4  6 12A12B13A···13F26A···26F39A···39L52A···52L78A···78L156A···156X
order1222346121213···1326···2639···3952···5278···78156···156
size117878222222···22···22···22···22···22···2

81 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D6D12D13D26D39D52D78D156
kernelD156C156D78C52C39C26C13C12C6C4C3C2C1
# reps11211126612121224

Matrix representation of D156 in GL2(𝔽157) generated by

11998
5942
,
3859
74119
G:=sub<GL(2,GF(157))| [119,59,98,42],[38,74,59,119] >;

D156 in GAP, Magma, Sage, TeX

D_{156}
% in TeX

G:=Group("D156");
// GroupNames label

G:=SmallGroup(312,39);
// by ID

G=gap.SmallGroup(312,39);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,61,26,323,7204]);
// Polycyclic

G:=Group<a,b|a^156=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D156 in TeX

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