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G = C2×C140order 280 = 23·5·7

Abelian group of type [2,140]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C140, SmallGroup(280,29)

Series: Derived Chief Lower central Upper central

C1 — C2×C140
C1C2C14C70C140 — 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
(1 216)(2 217)(3 218)(4 219)(5 220)(6 221)(7 222)(8 223)(9 224)(10 225)(11 226)(12 227)(13 228)(14 229)(15 230)(16 231)(17 232)(18 233)(19 234)(20 235)(21 236)(22 237)(23 238)(24 239)(25 240)(26 241)(27 242)(28 243)(29 244)(30 245)(31 246)(32 247)(33 248)(34 249)(35 250)(36 251)(37 252)(38 253)(39 254)(40 255)(41 256)(42 257)(43 258)(44 259)(45 260)(46 261)(47 262)(48 263)(49 264)(50 265)(51 266)(52 267)(53 268)(54 269)(55 270)(56 271)(57 272)(58 273)(59 274)(60 275)(61 276)(62 277)(63 278)(64 279)(65 280)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 171)(97 172)(98 173)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 193)(119 194)(120 195)(121 196)(122 197)(123 198)(124 199)(125 200)(126 201)(127 202)(128 203)(129 204)(130 205)(131 206)(132 207)(133 208)(134 209)(135 210)(136 211)(137 212)(138 213)(139 214)(140 215)
(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)| (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,225)(11,226)(12,227)(13,228)(14,229)(15,230)(16,231)(17,232)(18,233)(19,234)(20,235)(21,236)(22,237)(23,238)(24,239)(25,240)(26,241)(27,242)(28,243)(29,244)(30,245)(31,246)(32,247)(33,248)(34,249)(35,250)(36,251)(37,252)(38,253)(39,254)(40,255)(41,256)(42,257)(43,258)(44,259)(45,260)(46,261)(47,262)(48,263)(49,264)(50,265)(51,266)(52,267)(53,268)(54,269)(55,270)(56,271)(57,272)(58,273)(59,274)(60,275)(61,276)(62,277)(63,278)(64,279)(65,280)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215), (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( (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,225)(11,226)(12,227)(13,228)(14,229)(15,230)(16,231)(17,232)(18,233)(19,234)(20,235)(21,236)(22,237)(23,238)(24,239)(25,240)(26,241)(27,242)(28,243)(29,244)(30,245)(31,246)(32,247)(33,248)(34,249)(35,250)(36,251)(37,252)(38,253)(39,254)(40,255)(41,256)(42,257)(43,258)(44,259)(45,260)(46,261)(47,262)(48,263)(49,264)(50,265)(51,266)(52,267)(53,268)(54,269)(55,270)(56,271)(57,272)(58,273)(59,274)(60,275)(61,276)(62,277)(63,278)(64,279)(65,280)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(91,166)(92,167)(93,168)(94,169)(95,170)(96,171)(97,172)(98,173)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,183)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,193)(119,194)(120,195)(121,196)(122,197)(123,198)(124,199)(125,200)(126,201)(127,202)(128,203)(129,204)(130,205)(131,206)(132,207)(133,208)(134,209)(135,210)(136,211)(137,212)(138,213)(139,214)(140,215), (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([(1,216),(2,217),(3,218),(4,219),(5,220),(6,221),(7,222),(8,223),(9,224),(10,225),(11,226),(12,227),(13,228),(14,229),(15,230),(16,231),(17,232),(18,233),(19,234),(20,235),(21,236),(22,237),(23,238),(24,239),(25,240),(26,241),(27,242),(28,243),(29,244),(30,245),(31,246),(32,247),(33,248),(34,249),(35,250),(36,251),(37,252),(38,253),(39,254),(40,255),(41,256),(42,257),(43,258),(44,259),(45,260),(46,261),(47,262),(48,263),(49,264),(50,265),(51,266),(52,267),(53,268),(54,269),(55,270),(56,271),(57,272),(58,273),(59,274),(60,275),(61,276),(62,277),(63,278),(64,279),(65,280),(66,141),(67,142),(68,143),(69,144),(70,145),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(91,166),(92,167),(93,168),(94,169),(95,170),(96,171),(97,172),(98,173),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,183),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,193),(119,194),(120,195),(121,196),(122,197),(123,198),(124,199),(125,200),(126,201),(127,202),(128,203),(129,204),(130,205),(131,206),(132,207),(133,208),(134,209),(135,210),(136,211),(137,212),(138,213),(139,214),(140,215)], [(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 2A2B2C4A4B4C4D5A5B5C5D7A···7F10A···10L14A···14R20A···20P28A···28X35A···35X70A···70BT140A···140CR
order1222444455557···710···1014···1420···2028···2835···3570···70140···140
size1111111111111···11···11···11···11···11···11···11···1

280 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C5C7C10C10C14C14C20C28C35C70C70C140
kernelC2×C140C140C2×C70C70C2×C28C2×C20C28C2×C14C20C2×C10C14C10C2×C4C4C22C2
# reps12144684126162424482496

Matrix representation of C2×C140 in GL3(𝔽281) generated by

100
010
00280
,
21100
02450
00165
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

Export

Subgroup lattice of C2×C140 in TeX

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