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G = D186order 372 = 22·3·31

Dihedral group

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

Aliases: D186, C2×D93, C62⋊S3, C6⋊D31, C32D62, C312D6, C1861C2, C932C22, sometimes denoted D372 or Dih186 or Dih372, SmallGroup(372,14)

Series: Derived Chief Lower central Upper central

C1C93 — D186
C1C31C93D93 — D186
C93 — D186
C1C2

Generators and relations for D186
 G = < a,b | a186=b2=1, bab=a-1 >

93C2
93C2
93C22
31S3
31S3
3D31
3D31
31D6
3D62

Smallest permutation representation of D186
On 186 points
Generators in S186
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186)
(1 186)(2 185)(3 184)(4 183)(5 182)(6 181)(7 180)(8 179)(9 178)(10 177)(11 176)(12 175)(13 174)(14 173)(15 172)(16 171)(17 170)(18 169)(19 168)(20 167)(21 166)(22 165)(23 164)(24 163)(25 162)(26 161)(27 160)(28 159)(29 158)(30 157)(31 156)(32 155)(33 154)(34 153)(35 152)(36 151)(37 150)(38 149)(39 148)(40 147)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 136)(52 135)(53 134)(54 133)(55 132)(56 131)(57 130)(58 129)(59 128)(60 127)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 120)(68 119)(69 118)(70 117)(71 116)(72 115)(73 114)(74 113)(75 112)(76 111)(77 110)(78 109)(79 108)(80 107)(81 106)(82 105)(83 104)(84 103)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)

G:=sub<Sym(186)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186), (1,186)(2,185)(3,184)(4,183)(5,182)(6,181)(7,180)(8,179)(9,178)(10,177)(11,176)(12,175)(13,174)(14,173)(15,172)(16,171)(17,170)(18,169)(19,168)(20,167)(21,166)(22,165)(23,164)(24,163)(25,162)(26,161)(27,160)(28,159)(29,158)(30,157)(31,156)(32,155)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186), (1,186)(2,185)(3,184)(4,183)(5,182)(6,181)(7,180)(8,179)(9,178)(10,177)(11,176)(12,175)(13,174)(14,173)(15,172)(16,171)(17,170)(18,169)(19,168)(20,167)(21,166)(22,165)(23,164)(24,163)(25,162)(26,161)(27,160)(28,159)(29,158)(30,157)(31,156)(32,155)(33,154)(34,153)(35,152)(36,151)(37,150)(38,149)(39,148)(40,147)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,120)(68,119)(69,118)(70,117)(71,116)(72,115)(73,114)(74,113)(75,112)(76,111)(77,110)(78,109)(79,108)(80,107)(81,106)(82,105)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186)], [(1,186),(2,185),(3,184),(4,183),(5,182),(6,181),(7,180),(8,179),(9,178),(10,177),(11,176),(12,175),(13,174),(14,173),(15,172),(16,171),(17,170),(18,169),(19,168),(20,167),(21,166),(22,165),(23,164),(24,163),(25,162),(26,161),(27,160),(28,159),(29,158),(30,157),(31,156),(32,155),(33,154),(34,153),(35,152),(36,151),(37,150),(38,149),(39,148),(40,147),(41,146),(42,145),(43,144),(44,143),(45,142),(46,141),(47,140),(48,139),(49,138),(50,137),(51,136),(52,135),(53,134),(54,133),(55,132),(56,131),(57,130),(58,129),(59,128),(60,127),(61,126),(62,125),(63,124),(64,123),(65,122),(66,121),(67,120),(68,119),(69,118),(70,117),(71,116),(72,115),(73,114),(74,113),(75,112),(76,111),(77,110),(78,109),(79,108),(80,107),(81,106),(82,105),(83,104),(84,103),(85,102),(86,101),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94)])

96 conjugacy classes

class 1 2A2B2C 3  6 31A···31O62A···62O93A···93AD186A···186AD
order12223631···3162···6293···93186···186
size119393222···22···22···22···2

96 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D31D62D93D186
kernelD186D93C186C62C31C6C3C2C1
# reps1211115153030

Matrix representation of D186 in GL2(𝔽373) generated by

101173
295136
,
10116
295272
G:=sub<GL(2,GF(373))| [101,295,173,136],[101,295,16,272] >;

D186 in GAP, Magma, Sage, TeX

D_{186}
% in TeX

G:=Group("D186");
// GroupNames label

G:=SmallGroup(372,14);
// by ID

G=gap.SmallGroup(372,14);
# by ID

G:=PCGroup([4,-2,-2,-3,-31,98,5763]);
// Polycyclic

G:=Group<a,b|a^186=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D186 in TeX

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