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G = D213order 426 = 2·3·71

Dihedral group

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

Aliases: D213, C71⋊S3, C3⋊D71, C2131C2, sometimes denoted D426 or Dih213 or Dih426, SmallGroup(426,3)

Series: Derived Chief Lower central Upper central

C1C213 — D213
C1C71C213 — D213
C213 — D213
C1

Generators and relations for D213
 G = < a,b | a213=b2=1, bab=a-1 >

213C2
71S3
3D71

Smallest permutation representation of D213
On 213 points
Generators in S213
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213)
(1 213)(2 212)(3 211)(4 210)(5 209)(6 208)(7 207)(8 206)(9 205)(10 204)(11 203)(12 202)(13 201)(14 200)(15 199)(16 198)(17 197)(18 196)(19 195)(20 194)(21 193)(22 192)(23 191)(24 190)(25 189)(26 188)(27 187)(28 186)(29 185)(30 184)(31 183)(32 182)(33 181)(34 180)(35 179)(36 178)(37 177)(38 176)(39 175)(40 174)(41 173)(42 172)(43 171)(44 170)(45 169)(46 168)(47 167)(48 166)(49 165)(50 164)(51 163)(52 162)(53 161)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(61 153)(62 152)(63 151)(64 150)(65 149)(66 148)(67 147)(68 146)(69 145)(70 144)(71 143)(72 142)(73 141)(74 140)(75 139)(76 138)(77 137)(78 136)(79 135)(80 134)(81 133)(82 132)(83 131)(84 130)(85 129)(86 128)(87 127)(88 126)(89 125)(90 124)(91 123)(92 122)(93 121)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)

G:=sub<Sym(213)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213), (1,213)(2,212)(3,211)(4,210)(5,209)(6,208)(7,207)(8,206)(9,205)(10,204)(11,203)(12,202)(13,201)(14,200)(15,199)(16,198)(17,197)(18,196)(19,195)(20,194)(21,193)(22,192)(23,191)(24,190)(25,189)(26,188)(27,187)(28,186)(29,185)(30,184)(31,183)(32,182)(33,181)(34,180)(35,179)(36,178)(37,177)(38,176)(39,175)(40,174)(41,173)(42,172)(43,171)(44,170)(45,169)(46,168)(47,167)(48,166)(49,165)(50,164)(51,163)(52,162)(53,161)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,153)(62,152)(63,151)(64,150)(65,149)(66,148)(67,147)(68,146)(69,145)(70,144)(71,143)(72,142)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(81,133)(82,132)(83,131)(84,130)(85,129)(86,128)(87,127)(88,126)(89,125)(90,124)(91,123)(92,122)(93,121)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213), (1,213)(2,212)(3,211)(4,210)(5,209)(6,208)(7,207)(8,206)(9,205)(10,204)(11,203)(12,202)(13,201)(14,200)(15,199)(16,198)(17,197)(18,196)(19,195)(20,194)(21,193)(22,192)(23,191)(24,190)(25,189)(26,188)(27,187)(28,186)(29,185)(30,184)(31,183)(32,182)(33,181)(34,180)(35,179)(36,178)(37,177)(38,176)(39,175)(40,174)(41,173)(42,172)(43,171)(44,170)(45,169)(46,168)(47,167)(48,166)(49,165)(50,164)(51,163)(52,162)(53,161)(54,160)(55,159)(56,158)(57,157)(58,156)(59,155)(60,154)(61,153)(62,152)(63,151)(64,150)(65,149)(66,148)(67,147)(68,146)(69,145)(70,144)(71,143)(72,142)(73,141)(74,140)(75,139)(76,138)(77,137)(78,136)(79,135)(80,134)(81,133)(82,132)(83,131)(84,130)(85,129)(86,128)(87,127)(88,126)(89,125)(90,124)(91,123)(92,122)(93,121)(94,120)(95,119)(96,118)(97,117)(98,116)(99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)(106,108) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213)], [(1,213),(2,212),(3,211),(4,210),(5,209),(6,208),(7,207),(8,206),(9,205),(10,204),(11,203),(12,202),(13,201),(14,200),(15,199),(16,198),(17,197),(18,196),(19,195),(20,194),(21,193),(22,192),(23,191),(24,190),(25,189),(26,188),(27,187),(28,186),(29,185),(30,184),(31,183),(32,182),(33,181),(34,180),(35,179),(36,178),(37,177),(38,176),(39,175),(40,174),(41,173),(42,172),(43,171),(44,170),(45,169),(46,168),(47,167),(48,166),(49,165),(50,164),(51,163),(52,162),(53,161),(54,160),(55,159),(56,158),(57,157),(58,156),(59,155),(60,154),(61,153),(62,152),(63,151),(64,150),(65,149),(66,148),(67,147),(68,146),(69,145),(70,144),(71,143),(72,142),(73,141),(74,140),(75,139),(76,138),(77,137),(78,136),(79,135),(80,134),(81,133),(82,132),(83,131),(84,130),(85,129),(86,128),(87,127),(88,126),(89,125),(90,124),(91,123),(92,122),(93,121),(94,120),(95,119),(96,118),(97,117),(98,116),(99,115),(100,114),(101,113),(102,112),(103,111),(104,110),(105,109),(106,108)])

108 conjugacy classes

class 1  2  3 71A···71AI213A···213BR
order12371···71213···213
size121322···22···2

108 irreducible representations

dim11222
type+++++
imageC1C2S3D71D213
kernelD213C213C71C3C1
# reps1113570

Matrix representation of D213 in GL2(𝔽853) generated by

743135
843842
,
744712
175109
G:=sub<GL(2,GF(853))| [743,843,135,842],[744,175,712,109] >;

D213 in GAP, Magma, Sage, TeX

D_{213}
% in TeX

G:=Group("D213");
// GroupNames label

G:=SmallGroup(426,3);
// by ID

G=gap.SmallGroup(426,3);
# by ID

G:=PCGroup([3,-2,-3,-71,25,3782]);
// Polycyclic

G:=Group<a,b|a^213=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D213 in TeX

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