metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D188, C4⋊D47, C47⋊1D4, C188⋊1C2, D94⋊1C2, C2.4D94, C94.3C22, sometimes denoted D376 or Dih188 or Dih376, SmallGroup(376,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D188
G = < a,b | a188=b2=1, bab=a-1 >
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 | 2A | 2B | 2C | 4 | 47A | ··· | 47W | 94A | ··· | 94W | 188A | ··· | 188AT |
| order | 1 | 2 | 2 | 2 | 4 | 47 | ··· | 47 | 94 | ··· | 94 | 188 | ··· | 188 |
| size | 1 | 1 | 94 | 94 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
97 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D47 | D94 | D188 |
| kernel | D188 | C188 | D94 | C47 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 23 | 23 | 46 |
Matrix representation of D188 ►in GL2(𝔽941) generated by
| 99 | 265 |
| 676 | 859 |
| 99 | 265 |
| 833 | 842 |
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
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