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G = D188order 376 = 23·47

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D188, C4⋊D47, C471D4, C1881C2, D941C2, C2.4D94, C94.3C22, sometimes denoted D376 or Dih188 or Dih376, SmallGroup(376,5)

Series: Derived Chief Lower central Upper central

C1C94 — D188
C1C47C94D94 — D188
C47C94 — D188
C1C2C4

Generators and relations for D188
 G = < a,b | a188=b2=1, bab=a-1 >

94C2
94C2
47C22
47C22
2D47
2D47
47D4

Smallest permutation representation of D188
On 188 points
Generators in S188
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188)
(1 188)(2 187)(3 186)(4 185)(5 184)(6 183)(7 182)(8 181)(9 180)(10 179)(11 178)(12 177)(13 176)(14 175)(15 174)(16 173)(17 172)(18 171)(19 170)(20 169)(21 168)(22 167)(23 166)(24 165)(25 164)(26 163)(27 162)(28 161)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 148)(42 147)(43 146)(44 145)(45 144)(46 143)(47 142)(48 141)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 128)(62 127)(63 126)(64 125)(65 124)(66 123)(67 122)(68 121)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)

G:=sub<Sym(188)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188), (1,188)(2,187)(3,186)(4,185)(5,184)(6,183)(7,182)(8,181)(9,180)(10,179)(11,178)(12,177)(13,176)(14,175)(15,174)(16,173)(17,172)(18,171)(19,170)(20,169)(21,168)(22,167)(23,166)(24,165)(25,164)(26,163)(27,162)(28,161)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188), (1,188)(2,187)(3,186)(4,185)(5,184)(6,183)(7,182)(8,181)(9,180)(10,179)(11,178)(12,177)(13,176)(14,175)(15,174)(16,173)(17,172)(18,171)(19,170)(20,169)(21,168)(22,167)(23,166)(24,165)(25,164)(26,163)(27,162)(28,161)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188)], [(1,188),(2,187),(3,186),(4,185),(5,184),(6,183),(7,182),(8,181),(9,180),(10,179),(11,178),(12,177),(13,176),(14,175),(15,174),(16,173),(17,172),(18,171),(19,170),(20,169),(21,168),(22,167),(23,166),(24,165),(25,164),(26,163),(27,162),(28,161),(29,160),(30,159),(31,158),(32,157),(33,156),(34,155),(35,154),(36,153),(37,152),(38,151),(39,150),(40,149),(41,148),(42,147),(43,146),(44,145),(45,144),(46,143),(47,142),(48,141),(49,140),(50,139),(51,138),(52,137),(53,136),(54,135),(55,134),(56,133),(57,132),(58,131),(59,130),(60,129),(61,128),(62,127),(63,126),(64,125),(65,124),(66,123),(67,122),(68,121),(69,120),(70,119),(71,118),(72,117),(73,116),(74,115),(75,114),(76,113),(77,112),(78,111),(79,110),(80,109),(81,108),(82,107),(83,106),(84,105),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95)])

97 conjugacy classes

class 1 2A2B2C 4 47A···47W94A···94W188A···188AT
order1222447···4794···94188···188
size11949422···22···22···2

97 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D47D94D188
kernelD188C188D94C47C4C2C1
# reps1121232346

Matrix representation of D188 in GL2(𝔽941) generated by

99265
676859
,
99265
833842
G:=sub<GL(2,GF(941))| [99,676,265,859],[99,833,265,842] >;

D188 in GAP, Magma, Sage, TeX

D_{188}
% in TeX

G:=Group("D188");
// GroupNames label

G:=SmallGroup(376,5);
// by ID

G=gap.SmallGroup(376,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-47,49,21,5891]);
// Polycyclic

G:=Group<a,b|a^188=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D188 in TeX

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