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G = D189order 378 = 2·33·7

Dihedral group

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

Aliases: D189, C27⋊D7, C7⋊D27, C3.D63, C9.D21, C1891C2, C63.1S3, C21.1D9, sometimes denoted D378 or Dih189 or Dih378, SmallGroup(378,5)

Series: Derived Chief Lower central Upper central

C1C189 — D189
C1C3C9C63C189 — D189
C189 — D189
C1

Generators and relations for D189
 G = < a,b | a189=b2=1, bab=a-1 >

189C2
63S3
27D7
21D9
9D21
7D27
3D63

Smallest permutation representation of D189
On 189 points
Generators in S189
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189)
(2 189)(3 188)(4 187)(5 186)(6 185)(7 184)(8 183)(9 182)(10 181)(11 180)(12 179)(13 178)(14 177)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 168)(24 167)(25 166)(26 165)(27 164)(28 163)(29 162)(30 161)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 150)(42 149)(43 148)(44 147)(45 146)(46 145)(47 144)(48 143)(49 142)(50 141)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 130)(62 129)(63 128)(64 127)(65 126)(66 125)(67 124)(68 123)(69 122)(70 121)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)

G:=sub<Sym(189)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189), (2,189)(3,188)(4,187)(5,186)(6,185)(7,184)(8,183)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189), (2,189)(3,188)(4,187)(5,186)(6,185)(7,184)(8,183)(9,182)(10,181)(11,180)(12,179)(13,178)(14,177)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,150)(42,149)(43,148)(44,147)(45,146)(46,145)(47,144)(48,143)(49,142)(50,141)(51,140)(52,139)(53,138)(54,137)(55,136)(56,135)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)], [(2,189),(3,188),(4,187),(5,186),(6,185),(7,184),(8,183),(9,182),(10,181),(11,180),(12,179),(13,178),(14,177),(15,176),(16,175),(17,174),(18,173),(19,172),(20,171),(21,170),(22,169),(23,168),(24,167),(25,166),(26,165),(27,164),(28,163),(29,162),(30,161),(31,160),(32,159),(33,158),(34,157),(35,156),(36,155),(37,154),(38,153),(39,152),(40,151),(41,150),(42,149),(43,148),(44,147),(45,146),(46,145),(47,144),(48,143),(49,142),(50,141),(51,140),(52,139),(53,138),(54,137),(55,136),(56,135),(57,134),(58,133),(59,132),(60,131),(61,130),(62,129),(63,128),(64,127),(65,126),(66,125),(67,124),(68,123),(69,122),(70,121),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96)])

96 conjugacy classes

class 1  2  3 7A7B7C9A9B9C21A···21F27A···27I63A···63R189A···189BB
order12377799921···2127···2763···63189···189
size118922222222···22···22···22···2

96 irreducible representations

dim112222222
type+++++++++
imageC1C2S3D7D9D21D27D63D189
kernelD189C189C63C27C21C9C7C3C1
# reps11133691854

Matrix representation of D189 in GL2(𝔽379) generated by

25592
287163
,
10
378378
G:=sub<GL(2,GF(379))| [255,287,92,163],[1,378,0,378] >;

D189 in GAP, Magma, Sage, TeX

D_{189}
% in TeX

G:=Group("D189");
// GroupNames label

G:=SmallGroup(378,5);
// by ID

G=gap.SmallGroup(378,5);
# by ID

G:=PCGroup([5,-2,-3,-7,-3,-3,461,1056,542,4203,138,6304]);
// Polycyclic

G:=Group<a,b|a^189=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D189 in TeX

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