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G = D184order 368 = 24·23

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D184, C231D8, C81D23, C1841C2, D921C2, C4.9D46, C2.4D92, C46.2D4, C92.9C22, sometimes denoted D368 or Dih184 or Dih368, SmallGroup(368,6)

Series: Derived Chief Lower central Upper central

C1C92 — D184
C1C23C46C92D92 — D184
C23C46C92 — D184
C1C2C4C8

Generators and relations for D184
 G = < a,b | a184=b2=1, bab=a-1 >

92C2
92C2
46C22
46C22
4D23
4D23
23D4
23D4
2D46
2D46
23D8

Smallest permutation representation of D184
On 184 points
Generators in S184
(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)
(1 184)(2 183)(3 182)(4 181)(5 180)(6 179)(7 178)(8 177)(9 176)(10 175)(11 174)(12 173)(13 172)(14 171)(15 170)(16 169)(17 168)(18 167)(19 166)(20 165)(21 164)(22 163)(23 162)(24 161)(25 160)(26 159)(27 158)(28 157)(29 156)(30 155)(31 154)(32 153)(33 152)(34 151)(35 150)(36 149)(37 148)(38 147)(39 146)(40 145)(41 144)(42 143)(43 142)(44 141)(45 140)(46 139)(47 138)(48 137)(49 136)(50 135)(51 134)(52 133)(53 132)(54 131)(55 130)(56 129)(57 128)(58 127)(59 126)(60 125)(61 124)(62 123)(63 122)(64 121)(65 120)(66 119)(67 118)(68 117)(69 116)(70 115)(71 114)(72 113)(73 112)(74 111)(75 110)(76 109)(77 108)(78 107)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)

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

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

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

95 conjugacy classes

class 1 2A2B2C 4 8A8B23A···23K46A···46K92A···92V184A···184AR
order122248823···2346···4692···92184···184
size1192922222···22···22···22···2

95 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D4D8D23D46D92D184
kernelD184C184D92C46C23C8C4C2C1
# reps1121211112244

Matrix representation of D184 in GL2(𝔽1289) generated by

1182397
7151178
,
1102790
1248187
G:=sub<GL(2,GF(1289))| [1182,715,397,1178],[1102,1248,790,187] >;

D184 in GAP, Magma, Sage, TeX

D_{184}
% in TeX

G:=Group("D184");
// GroupNames label

G:=SmallGroup(368,6);
// by ID

G=gap.SmallGroup(368,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-23,61,66,182,42,8804]);
// Polycyclic

G:=Group<a,b|a^184=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D184 in TeX

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