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G = D177order 354 = 2·3·59

Dihedral group

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

Aliases: D177, C59⋊S3, C3⋊D59, C1771C2, sometimes denoted D354 or Dih177 or Dih354, SmallGroup(354,3)

Series: Derived Chief Lower central Upper central

C1C177 — D177
C1C59C177 — D177
C177 — D177
C1

Generators and relations for D177
 G = < a,b | a177=b2=1, bab=a-1 >

177C2
59S3
3D59

Smallest permutation representation of D177
On 177 points
Generators in S177
(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)
(1 177)(2 176)(3 175)(4 174)(5 173)(6 172)(7 171)(8 170)(9 169)(10 168)(11 167)(12 166)(13 165)(14 164)(15 163)(16 162)(17 161)(18 160)(19 159)(20 158)(21 157)(22 156)(23 155)(24 154)(25 153)(26 152)(27 151)(28 150)(29 149)(30 148)(31 147)(32 146)(33 145)(34 144)(35 143)(36 142)(37 141)(38 140)(39 139)(40 138)(41 137)(42 136)(43 135)(44 134)(45 133)(46 132)(47 131)(48 130)(49 129)(50 128)(51 127)(52 126)(53 125)(54 124)(55 123)(56 122)(57 121)(58 120)(59 119)(60 118)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 90)

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

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

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

90 conjugacy classes

class 1  2  3 59A···59AC177A···177BF
order12359···59177···177
size117722···22···2

90 irreducible representations

dim11222
type+++++
imageC1C2S3D59D177
kernelD177C177C59C3C1
# reps1112958

Matrix representation of D177 in GL2(𝔽709) generated by

388460
249308
,
388460
485321
G:=sub<GL(2,GF(709))| [388,249,460,308],[388,485,460,321] >;

D177 in GAP, Magma, Sage, TeX

D_{177}
% in TeX

G:=Group("D177");
// GroupNames label

G:=SmallGroup(354,3);
// by ID

G=gap.SmallGroup(354,3);
# by ID

G:=PCGroup([3,-2,-3,-59,25,3134]);
// Polycyclic

G:=Group<a,b|a^177=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D177 in TeX

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