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G = D178order 356 = 22·89

Dihedral group

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

Aliases: D178, C2×D89, C178⋊C2, C89⋊C22, sometimes denoted D356 or Dih178 or Dih356, SmallGroup(356,4)

Series: Derived Chief Lower central Upper central

C1C89 — D178
C1C89D89 — D178
C89 — D178
C1C2

Generators and relations for D178
 G = < a,b | a178=b2=1, bab=a-1 >

89C2
89C2
89C22

Smallest permutation representation of D178
On 178 points
Generators in S178
(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)
(1 178)(2 177)(3 176)(4 175)(5 174)(6 173)(7 172)(8 171)(9 170)(10 169)(11 168)(12 167)(13 166)(14 165)(15 164)(16 163)(17 162)(18 161)(19 160)(20 159)(21 158)(22 157)(23 156)(24 155)(25 154)(26 153)(27 152)(28 151)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 140)(40 139)(41 138)(42 137)(43 136)(44 135)(45 134)(46 133)(47 132)(48 131)(49 130)(50 129)(51 128)(52 127)(53 126)(54 125)(55 124)(56 123)(57 122)(58 121)(59 120)(60 119)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 100)(80 99)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)

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

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

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

92 conjugacy classes

class 1 2A2B2C89A···89AR178A···178AR
order122289···89178···178
size1189892···22···2

92 irreducible representations

dim11122
type+++++
imageC1C2C2D89D178
kernelD178D89C178C2C1
# reps1214444

Matrix representation of D178 in GL2(𝔽179) generated by

57176
33
,
57176
128122
G:=sub<GL(2,GF(179))| [57,3,176,3],[57,128,176,122] >;

D178 in GAP, Magma, Sage, TeX

D_{178}
% in TeX

G:=Group("D178");
// GroupNames label

G:=SmallGroup(356,4);
// by ID

G=gap.SmallGroup(356,4);
# by ID

G:=PCGroup([3,-2,-2,-89,3170]);
// Polycyclic

G:=Group<a,b|a^178=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D178 in TeX

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