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G = C2×C138order 276 = 22·3·23

Abelian group of type [2,138]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C138, SmallGroup(276,10)

Series: Derived Chief Lower central Upper central

C1 — C2×C138
C1C23C69C138 — C2×C138
C1 — C2×C138
C1 — C2×C138

Generators and relations for C2×C138
 G = < a,b | a2=b138=1, ab=ba >


Smallest permutation representation of C2×C138
Regular action on 276 points
Generators in S276
(1 236)(2 237)(3 238)(4 239)(5 240)(6 241)(7 242)(8 243)(9 244)(10 245)(11 246)(12 247)(13 248)(14 249)(15 250)(16 251)(17 252)(18 253)(19 254)(20 255)(21 256)(22 257)(23 258)(24 259)(25 260)(26 261)(27 262)(28 263)(29 264)(30 265)(31 266)(32 267)(33 268)(34 269)(35 270)(36 271)(37 272)(38 273)(39 274)(40 275)(41 276)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 179)(83 180)(84 181)(85 182)(86 183)(87 184)(88 185)(89 186)(90 187)(91 188)(92 189)(93 190)(94 191)(95 192)(96 193)(97 194)(98 195)(99 196)(100 197)(101 198)(102 199)(103 200)(104 201)(105 202)(106 203)(107 204)(108 205)(109 206)(110 207)(111 208)(112 209)(113 210)(114 211)(115 212)(116 213)(117 214)(118 215)(119 216)(120 217)(121 218)(122 219)(123 220)(124 221)(125 222)(126 223)(127 224)(128 225)(129 226)(130 227)(131 228)(132 229)(133 230)(134 231)(135 232)(136 233)(137 234)(138 235)
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276)

G:=sub<Sym(276)| (1,236)(2,237)(3,238)(4,239)(5,240)(6,241)(7,242)(8,243)(9,244)(10,245)(11,246)(12,247)(13,248)(14,249)(15,250)(16,251)(17,252)(18,253)(19,254)(20,255)(21,256)(22,257)(23,258)(24,259)(25,260)(26,261)(27,262)(28,263)(29,264)(30,265)(31,266)(32,267)(33,268)(34,269)(35,270)(36,271)(37,272)(38,273)(39,274)(40,275)(41,276)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,178)(82,179)(83,180)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201)(105,202)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,209)(113,210)(114,211)(115,212)(116,213)(117,214)(118,215)(119,216)(120,217)(121,218)(122,219)(123,220)(124,221)(125,222)(126,223)(127,224)(128,225)(129,226)(130,227)(131,228)(132,229)(133,230)(134,231)(135,232)(136,233)(137,234)(138,235), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276)>;

G:=Group( (1,236)(2,237)(3,238)(4,239)(5,240)(6,241)(7,242)(8,243)(9,244)(10,245)(11,246)(12,247)(13,248)(14,249)(15,250)(16,251)(17,252)(18,253)(19,254)(20,255)(21,256)(22,257)(23,258)(24,259)(25,260)(26,261)(27,262)(28,263)(29,264)(30,265)(31,266)(32,267)(33,268)(34,269)(35,270)(36,271)(37,272)(38,273)(39,274)(40,275)(41,276)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,178)(82,179)(83,180)(84,181)(85,182)(86,183)(87,184)(88,185)(89,186)(90,187)(91,188)(92,189)(93,190)(94,191)(95,192)(96,193)(97,194)(98,195)(99,196)(100,197)(101,198)(102,199)(103,200)(104,201)(105,202)(106,203)(107,204)(108,205)(109,206)(110,207)(111,208)(112,209)(113,210)(114,211)(115,212)(116,213)(117,214)(118,215)(119,216)(120,217)(121,218)(122,219)(123,220)(124,221)(125,222)(126,223)(127,224)(128,225)(129,226)(130,227)(131,228)(132,229)(133,230)(134,231)(135,232)(136,233)(137,234)(138,235), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276) );

G=PermutationGroup([(1,236),(2,237),(3,238),(4,239),(5,240),(6,241),(7,242),(8,243),(9,244),(10,245),(11,246),(12,247),(13,248),(14,249),(15,250),(16,251),(17,252),(18,253),(19,254),(20,255),(21,256),(22,257),(23,258),(24,259),(25,260),(26,261),(27,262),(28,263),(29,264),(30,265),(31,266),(32,267),(33,268),(34,269),(35,270),(36,271),(37,272),(38,273),(39,274),(40,275),(41,276),(42,139),(43,140),(44,141),(45,142),(46,143),(47,144),(48,145),(49,146),(50,147),(51,148),(52,149),(53,150),(54,151),(55,152),(56,153),(57,154),(58,155),(59,156),(60,157),(61,158),(62,159),(63,160),(64,161),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,169),(73,170),(74,171),(75,172),(76,173),(77,174),(78,175),(79,176),(80,177),(81,178),(82,179),(83,180),(84,181),(85,182),(86,183),(87,184),(88,185),(89,186),(90,187),(91,188),(92,189),(93,190),(94,191),(95,192),(96,193),(97,194),(98,195),(99,196),(100,197),(101,198),(102,199),(103,200),(104,201),(105,202),(106,203),(107,204),(108,205),(109,206),(110,207),(111,208),(112,209),(113,210),(114,211),(115,212),(116,213),(117,214),(118,215),(119,216),(120,217),(121,218),(122,219),(123,220),(124,221),(125,222),(126,223),(127,224),(128,225),(129,226),(130,227),(131,228),(132,229),(133,230),(134,231),(135,232),(136,233),(137,234),(138,235)], [(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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276)])

276 conjugacy classes

class 1 2A2B2C3A3B6A···6F23A···23V46A···46BN69A···69AR138A···138EB
order1222336···623···2346···4669···69138···138
size1111111···11···11···11···11···1

276 irreducible representations

dim11111111
type++
imageC1C2C3C6C23C46C69C138
kernelC2×C138C138C2×C46C46C2×C6C6C22C2
# reps1326226644132

Matrix representation of C2×C138 in GL2(𝔽139) generated by

1380
01
,
370
0138
G:=sub<GL(2,GF(139))| [138,0,0,1],[37,0,0,138] >;

C2×C138 in GAP, Magma, Sage, TeX

C_2\times C_{138}
% in TeX

G:=Group("C2xC138");
// GroupNames label

G:=SmallGroup(276,10);
// by ID

G=gap.SmallGroup(276,10);
# by ID

G:=PCGroup([4,-2,-2,-3,-23]);
// Polycyclic

G:=Group<a,b|a^2=b^138=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C138 in TeX

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