direct product, abelian, monomial, 2-elementary
Aliases: C2×C140, SmallGroup(280,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C140 |
| C1 — C2×C140 |
| C1 — C2×C140 |
Generators and relations for C2×C140
G = < a,b | a2=b140=1, ab=ba >
Smallest permutation representation of C2×C140
►Regular action on 280 points
Generators in S280
(1 158)(2 159)(3 160)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 168)(12 169)(13 170)(14 171)(15 172)(16 173)(17 174)(18 175)(19 176)(20 177)(21 178)(22 179)(23 180)(24 181)(25 182)(26 183)(27 184)(28 185)(29 186)(30 187)(31 188)(32 189)(33 190)(34 191)(35 192)(36 193)(37 194)(38 195)(39 196)(40 197)(41 198)(42 199)(43 200)(44 201)(45 202)(46 203)(47 204)(48 205)(49 206)(50 207)(51 208)(52 209)(53 210)(54 211)(55 212)(56 213)(57 214)(58 215)(59 216)(60 217)(61 218)(62 219)(63 220)(64 221)(65 222)(66 223)(67 224)(68 225)(69 226)(70 227)(71 228)(72 229)(73 230)(74 231)(75 232)(76 233)(77 234)(78 235)(79 236)(80 237)(81 238)(82 239)(83 240)(84 241)(85 242)(86 243)(87 244)(88 245)(89 246)(90 247)(91 248)(92 249)(93 250)(94 251)(95 252)(96 253)(97 254)(98 255)(99 256)(100 257)(101 258)(102 259)(103 260)(104 261)(105 262)(106 263)(107 264)(108 265)(109 266)(110 267)(111 268)(112 269)(113 270)(114 271)(115 272)(116 273)(117 274)(118 275)(119 276)(120 277)(121 278)(122 279)(123 280)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)
G:=sub<Sym(280)| (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,182)(26,183)(27,184)(28,185)(29,186)(30,187)(31,188)(32,189)(33,190)(34,191)(35,192)(36,193)(37,194)(38,195)(39,196)(40,197)(41,198)(42,199)(43,200)(44,201)(45,202)(46,203)(47,204)(48,205)(49,206)(50,207)(51,208)(52,209)(53,210)(54,211)(55,212)(56,213)(57,214)(58,215)(59,216)(60,217)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,229)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,241)(85,242)(86,243)(87,244)(88,245)(89,246)(90,247)(91,248)(92,249)(93,250)(94,251)(95,252)(96,253)(97,254)(98,255)(99,256)(100,257)(101,258)(102,259)(103,260)(104,261)(105,262)(106,263)(107,264)(108,265)(109,266)(110,267)(111,268)(112,269)(113,270)(114,271)(115,272)(116,273)(117,274)(118,275)(119,276)(120,277)(121,278)(122,279)(123,280)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)>;
G:=Group( (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,182)(26,183)(27,184)(28,185)(29,186)(30,187)(31,188)(32,189)(33,190)(34,191)(35,192)(36,193)(37,194)(38,195)(39,196)(40,197)(41,198)(42,199)(43,200)(44,201)(45,202)(46,203)(47,204)(48,205)(49,206)(50,207)(51,208)(52,209)(53,210)(54,211)(55,212)(56,213)(57,214)(58,215)(59,216)(60,217)(61,218)(62,219)(63,220)(64,221)(65,222)(66,223)(67,224)(68,225)(69,226)(70,227)(71,228)(72,229)(73,230)(74,231)(75,232)(76,233)(77,234)(78,235)(79,236)(80,237)(81,238)(82,239)(83,240)(84,241)(85,242)(86,243)(87,244)(88,245)(89,246)(90,247)(91,248)(92,249)(93,250)(94,251)(95,252)(96,253)(97,254)(98,255)(99,256)(100,257)(101,258)(102,259)(103,260)(104,261)(105,262)(106,263)(107,264)(108,265)(109,266)(110,267)(111,268)(112,269)(113,270)(114,271)(115,272)(116,273)(117,274)(118,275)(119,276)(120,277)(121,278)(122,279)(123,280)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280) );
G=PermutationGroup([[(1,158),(2,159),(3,160),(4,161),(5,162),(6,163),(7,164),(8,165),(9,166),(10,167),(11,168),(12,169),(13,170),(14,171),(15,172),(16,173),(17,174),(18,175),(19,176),(20,177),(21,178),(22,179),(23,180),(24,181),(25,182),(26,183),(27,184),(28,185),(29,186),(30,187),(31,188),(32,189),(33,190),(34,191),(35,192),(36,193),(37,194),(38,195),(39,196),(40,197),(41,198),(42,199),(43,200),(44,201),(45,202),(46,203),(47,204),(48,205),(49,206),(50,207),(51,208),(52,209),(53,210),(54,211),(55,212),(56,213),(57,214),(58,215),(59,216),(60,217),(61,218),(62,219),(63,220),(64,221),(65,222),(66,223),(67,224),(68,225),(69,226),(70,227),(71,228),(72,229),(73,230),(74,231),(75,232),(76,233),(77,234),(78,235),(79,236),(80,237),(81,238),(82,239),(83,240),(84,241),(85,242),(86,243),(87,244),(88,245),(89,246),(90,247),(91,248),(92,249),(93,250),(94,251),(95,252),(96,253),(97,254),(98,255),(99,256),(100,257),(101,258),(102,259),(103,260),(104,261),(105,262),(106,263),(107,264),(108,265),(109,266),(110,267),(111,268),(112,269),(113,270),(114,271),(115,272),(116,273),(117,274),(118,275),(119,276),(120,277),(121,278),(122,279),(123,280),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157)], [(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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | ··· | 10L | 14A | ··· | 14R | 20A | ··· | 20P | 28A | ··· | 28X | 35A | ··· | 35X | 70A | ··· | 70BT | 140A | ··· | 140CR |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 20 | ··· | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C7 | C10 | C10 | C14 | C14 | C20 | C28 | C35 | C70 | C70 | C140 |
| kernel | C2×C140 | C140 | C2×C70 | C70 | C2×C28 | C2×C20 | C28 | C2×C14 | C20 | C2×C10 | C14 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 6 | 8 | 4 | 12 | 6 | 16 | 24 | 24 | 48 | 24 | 96 |
Matrix representation of C2×C140 ►in GL3(𝔽281) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 280 |
| 211 | 0 | 0 |
| 0 | 245 | 0 |
| 0 | 0 | 165 |
G:=sub<GL(3,GF(281))| [1,0,0,0,1,0,0,0,280],[211,0,0,0,245,0,0,0,165] >;
C2×C140 in GAP, Magma, Sage, TeX
C_2\times C_{140} % in TeX
G:=Group("C2xC140"); // GroupNames label
G:=SmallGroup(280,29);
// by ID
G=gap.SmallGroup(280,29);
# by ID
G:=PCGroup([5,-2,-2,-5,-7,-2,700]);
// Polycyclic
G:=Group<a,b|a^2=b^140=1,a*b=b*a>;
// generators/relations
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