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G = D179order 358 = 2·179

Dihedral group

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

Aliases: D179, C179⋊C2, sometimes denoted D358 or Dih179 or Dih358, SmallGroup(358,1)

Series: Derived Chief Lower central Upper central

C1C179 — D179
C1C179 — D179
C179 — D179
C1

Generators and relations for D179
 G = < a,b | a179=b2=1, bab=a-1 >

179C2

Smallest permutation representation of D179
On 179 points: primitive
Generators in S179
(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)
(1 179)(2 178)(3 177)(4 176)(5 175)(6 174)(7 173)(8 172)(9 171)(10 170)(11 169)(12 168)(13 167)(14 166)(15 165)(16 164)(17 163)(18 162)(19 161)(20 160)(21 159)(22 158)(23 157)(24 156)(25 155)(26 154)(27 153)(28 152)(29 151)(30 150)(31 149)(32 148)(33 147)(34 146)(35 145)(36 144)(37 143)(38 142)(39 141)(40 140)(41 139)(42 138)(43 137)(44 136)(45 135)(46 134)(47 133)(48 132)(49 131)(50 130)(51 129)(52 128)(53 127)(54 126)(55 125)(56 124)(57 123)(58 122)(59 121)(60 120)(61 119)(62 118)(63 117)(64 116)(65 115)(66 114)(67 113)(68 112)(69 111)(70 110)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 100)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)

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

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

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

91 conjugacy classes

class 1  2 179A···179CK
order12179···179
size11792···2

91 irreducible representations

dim112
type+++
imageC1C2D179
kernelD179C179C1
# reps1189

Matrix representation of D179 in GL2(𝔽359) generated by

34358
10
,
34358
78325
G:=sub<GL(2,GF(359))| [34,1,358,0],[34,78,358,325] >;

D179 in GAP, Magma, Sage, TeX

D_{179}
% in TeX

G:=Group("D179");
// GroupNames label

G:=SmallGroup(358,1);
// by ID

G=gap.SmallGroup(358,1);
# by ID

G:=PCGroup([2,-2,-179,1425]);
// Polycyclic

G:=Group<a,b|a^179=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D179 in TeX

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