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G = D249order 498 = 2·3·83

Dihedral group

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

Aliases: D249, C83⋊S3, C3⋊D83, C2491C2, sometimes denoted D498 or Dih249 or Dih498, SmallGroup(498,3)

Series: Derived Chief Lower central Upper central

C1C249 — D249
C1C83C249 — D249
C249 — D249
C1

Generators and relations for D249
 G = < a,b | a249=b2=1, bab=a-1 >

249C2
83S3
3D83

Smallest permutation representation of D249
On 249 points
Generators in S249
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249)
(1 249)(2 248)(3 247)(4 246)(5 245)(6 244)(7 243)(8 242)(9 241)(10 240)(11 239)(12 238)(13 237)(14 236)(15 235)(16 234)(17 233)(18 232)(19 231)(20 230)(21 229)(22 228)(23 227)(24 226)(25 225)(26 224)(27 223)(28 222)(29 221)(30 220)(31 219)(32 218)(33 217)(34 216)(35 215)(36 214)(37 213)(38 212)(39 211)(40 210)(41 209)(42 208)(43 207)(44 206)(45 205)(46 204)(47 203)(48 202)(49 201)(50 200)(51 199)(52 198)(53 197)(54 196)(55 195)(56 194)(57 193)(58 192)(59 191)(60 190)(61 189)(62 188)(63 187)(64 186)(65 185)(66 184)(67 183)(68 182)(69 181)(70 180)(71 179)(72 178)(73 177)(74 176)(75 175)(76 174)(77 173)(78 172)(79 171)(80 170)(81 169)(82 168)(83 167)(84 166)(85 165)(86 164)(87 163)(88 162)(89 161)(90 160)(91 159)(92 158)(93 157)(94 156)(95 155)(96 154)(97 153)(98 152)(99 151)(100 150)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 140)(111 139)(112 138)(113 137)(114 136)(115 135)(116 134)(117 133)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)

G:=sub<Sym(249)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,249)(2,248)(3,247)(4,246)(5,245)(6,244)(7,243)(8,242)(9,241)(10,240)(11,239)(12,238)(13,237)(14,236)(15,235)(16,234)(17,233)(18,232)(19,231)(20,230)(21,229)(22,228)(23,227)(24,226)(25,225)(26,224)(27,223)(28,222)(29,221)(30,220)(31,219)(32,218)(33,217)(34,216)(35,215)(36,214)(37,213)(38,212)(39,211)(40,210)(41,209)(42,208)(43,207)(44,206)(45,205)(46,204)(47,203)(48,202)(49,201)(50,200)(51,199)(52,198)(53,197)(54,196)(55,195)(56,194)(57,193)(58,192)(59,191)(60,190)(61,189)(62,188)(63,187)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,161)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,140)(111,139)(112,138)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249), (1,249)(2,248)(3,247)(4,246)(5,245)(6,244)(7,243)(8,242)(9,241)(10,240)(11,239)(12,238)(13,237)(14,236)(15,235)(16,234)(17,233)(18,232)(19,231)(20,230)(21,229)(22,228)(23,227)(24,226)(25,225)(26,224)(27,223)(28,222)(29,221)(30,220)(31,219)(32,218)(33,217)(34,216)(35,215)(36,214)(37,213)(38,212)(39,211)(40,210)(41,209)(42,208)(43,207)(44,206)(45,205)(46,204)(47,203)(48,202)(49,201)(50,200)(51,199)(52,198)(53,197)(54,196)(55,195)(56,194)(57,193)(58,192)(59,191)(60,190)(61,189)(62,188)(63,187)(64,186)(65,185)(66,184)(67,183)(68,182)(69,181)(70,180)(71,179)(72,178)(73,177)(74,176)(75,175)(76,174)(77,173)(78,172)(79,171)(80,170)(81,169)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,161)(90,160)(91,159)(92,158)(93,157)(94,156)(95,155)(96,154)(97,153)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,140)(111,139)(112,138)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249)], [(1,249),(2,248),(3,247),(4,246),(5,245),(6,244),(7,243),(8,242),(9,241),(10,240),(11,239),(12,238),(13,237),(14,236),(15,235),(16,234),(17,233),(18,232),(19,231),(20,230),(21,229),(22,228),(23,227),(24,226),(25,225),(26,224),(27,223),(28,222),(29,221),(30,220),(31,219),(32,218),(33,217),(34,216),(35,215),(36,214),(37,213),(38,212),(39,211),(40,210),(41,209),(42,208),(43,207),(44,206),(45,205),(46,204),(47,203),(48,202),(49,201),(50,200),(51,199),(52,198),(53,197),(54,196),(55,195),(56,194),(57,193),(58,192),(59,191),(60,190),(61,189),(62,188),(63,187),(64,186),(65,185),(66,184),(67,183),(68,182),(69,181),(70,180),(71,179),(72,178),(73,177),(74,176),(75,175),(76,174),(77,173),(78,172),(79,171),(80,170),(81,169),(82,168),(83,167),(84,166),(85,165),(86,164),(87,163),(88,162),(89,161),(90,160),(91,159),(92,158),(93,157),(94,156),(95,155),(96,154),(97,153),(98,152),(99,151),(100,150),(101,149),(102,148),(103,147),(104,146),(105,145),(106,144),(107,143),(108,142),(109,141),(110,140),(111,139),(112,138),(113,137),(114,136),(115,135),(116,134),(117,133),(118,132),(119,131),(120,130),(121,129),(122,128),(123,127),(124,126)])

126 conjugacy classes

class 1  2  3 83A···83AO249A···249CD
order12383···83249···249
size124922···22···2

126 irreducible representations

dim11222
type+++++
imageC1C2S3D83D249
kernelD249C249C83C3C1
# reps1114182

Matrix representation of D249 in GL2(𝔽499) generated by

201461
456455
,
217491
397282
G:=sub<GL(2,GF(499))| [201,456,461,455],[217,397,491,282] >;

D249 in GAP, Magma, Sage, TeX

D_{249}
% in TeX

G:=Group("D249");
// GroupNames label

G:=SmallGroup(498,3);
// by ID

G=gap.SmallGroup(498,3);
# by ID

G:=PCGroup([3,-2,-3,-83,25,4430]);
// Polycyclic

G:=Group<a,b|a^249=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D249 in TeX

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