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G = D174order 348 = 22·3·29

Dihedral group

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

Aliases: D174, C2×D87, C58⋊S3, C6⋊D29, C32D58, C292D6, C1741C2, C872C22, sometimes denoted D348 or Dih174 or Dih348, SmallGroup(348,11)

Series: Derived Chief Lower central Upper central

C1C87 — D174
C1C29C87D87 — D174
C87 — D174
C1C2

Generators and relations for D174
 G = < a,b | a174=b2=1, bab=a-1 >

87C2
87C2
87C22
29S3
29S3
3D29
3D29
29D6
3D58

Smallest permutation representation of D174
On 174 points
Generators in S174
(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)
(1 174)(2 173)(3 172)(4 171)(5 170)(6 169)(7 168)(8 167)(9 166)(10 165)(11 164)(12 163)(13 162)(14 161)(15 160)(16 159)(17 158)(18 157)(19 156)(20 155)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 140)(36 139)(37 138)(38 137)(39 136)(40 135)(41 134)(42 133)(43 132)(44 131)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 114)(62 113)(63 112)(64 111)(65 110)(66 109)(67 108)(68 107)(69 106)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)

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

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

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

90 conjugacy classes

class 1 2A2B2C 3  6 29A···29N58A···58N87A···87AB174A···174AB
order12223629···2958···5887···87174···174
size118787222···22···22···22···2

90 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D29D58D87D174
kernelD174D87C174C58C29C6C3C2C1
# reps1211114142828

Matrix representation of D174 in GL3(𝔽349) generated by

34800
015992
0207111
,
100
0211217
020138
G:=sub<GL(3,GF(349))| [348,0,0,0,159,207,0,92,111],[1,0,0,0,211,20,0,217,138] >;

D174 in GAP, Magma, Sage, TeX

D_{174}
% in TeX

G:=Group("D174");
// GroupNames label

G:=SmallGroup(348,11);
// by ID

G=gap.SmallGroup(348,11);
# by ID

G:=PCGroup([4,-2,-2,-3,-29,98,5379]);
// Polycyclic

G:=Group<a,b|a^174=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D174 in TeX

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