direct product, abelian, monomial, 2-elementary
Aliases: C22×C70, SmallGroup(280,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C70 |
| C1 — C22×C70 |
| C1 — C22×C70 |
Generators and relations for C22×C70
G = < a,b,c | a2=b2=c70=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C22, C5, C7, C23, C10, C14, C2×C10, C2×C14, C35, C22×C10, C22×C14, C70, C2×C70, C22×C70
Quotients: C1, C2, C22, C5, C7, C23, C10, C14, C2×C10, C2×C14, C35, C22×C10, C22×C14, C70, C2×C70, C22×C70
►Regular action on 280 points
Generators in S280
(1 192)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 200)(10 201)(11 202)(12 203)(13 204)(14 205)(15 206)(16 207)(17 208)(18 209)(19 210)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 181)(61 182)(62 183)(63 184)(64 185)(65 186)(66 187)(67 188)(68 189)(69 190)(70 191)(71 273)(72 274)(73 275)(74 276)(75 277)(76 278)(77 279)(78 280)(79 211)(80 212)(81 213)(82 214)(83 215)(84 216)(85 217)(86 218)(87 219)(88 220)(89 221)(90 222)(91 223)(92 224)(93 225)(94 226)(95 227)(96 228)(97 229)(98 230)(99 231)(100 232)(101 233)(102 234)(103 235)(104 236)(105 237)(106 238)(107 239)(108 240)(109 241)(110 242)(111 243)(112 244)(113 245)(114 246)(115 247)(116 248)(117 249)(118 250)(119 251)(120 252)(121 253)(122 254)(123 255)(124 256)(125 257)(126 258)(127 259)(128 260)(129 261)(130 262)(131 263)(132 264)(133 265)(134 266)(135 267)(136 268)(137 269)(138 270)(139 271)(140 272)
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 225)(155 226)(156 227)(157 228)(158 229)(159 230)(160 231)(161 232)(162 233)(163 234)(164 235)(165 236)(166 237)(167 238)(168 239)(169 240)(170 241)(171 242)(172 243)(173 244)(174 245)(175 246)(176 247)(177 248)(178 249)(179 250)(180 251)(181 252)(182 253)(183 254)(184 255)(185 256)(186 257)(187 258)(188 259)(189 260)(190 261)(191 262)(192 263)(193 264)(194 265)(195 266)(196 267)(197 268)(198 269)(199 270)(200 271)(201 272)(202 273)(203 274)(204 275)(205 276)(206 277)(207 278)(208 279)(209 280)(210 211)
(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)| (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,208)(18,209)(19,210)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,273)(72,274)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,211)(80,212)(81,213)(82,214)(83,215)(84,216)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,225)(94,226)(95,227)(96,228)(97,229)(98,230)(99,231)(100,232)(101,233)(102,234)(103,235)(104,236)(105,237)(106,238)(107,239)(108,240)(109,241)(110,242)(111,243)(112,244)(113,245)(114,246)(115,247)(116,248)(117,249)(118,250)(119,251)(120,252)(121,253)(122,254)(123,255)(124,256)(125,257)(126,258)(127,259)(128,260)(129,261)(130,262)(131,263)(132,264)(133,265)(134,266)(135,267)(136,268)(137,269)(138,270)(139,271)(140,272), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,225)(155,226)(156,227)(157,228)(158,229)(159,230)(160,231)(161,232)(162,233)(163,234)(164,235)(165,236)(166,237)(167,238)(168,239)(169,240)(170,241)(171,242)(172,243)(173,244)(174,245)(175,246)(176,247)(177,248)(178,249)(179,250)(180,251)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,261)(191,262)(192,263)(193,264)(194,265)(195,266)(196,267)(197,268)(198,269)(199,270)(200,271)(201,272)(202,273)(203,274)(204,275)(205,276)(206,277)(207,278)(208,279)(209,280)(210,211), (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( (1,192)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,200)(10,201)(11,202)(12,203)(13,204)(14,205)(15,206)(16,207)(17,208)(18,209)(19,210)(20,141)(21,142)(22,143)(23,144)(24,145)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,153)(33,154)(34,155)(35,156)(36,157)(37,158)(38,159)(39,160)(40,161)(41,162)(42,163)(43,164)(44,165)(45,166)(46,167)(47,168)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,178)(58,179)(59,180)(60,181)(61,182)(62,183)(63,184)(64,185)(65,186)(66,187)(67,188)(68,189)(69,190)(70,191)(71,273)(72,274)(73,275)(74,276)(75,277)(76,278)(77,279)(78,280)(79,211)(80,212)(81,213)(82,214)(83,215)(84,216)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,225)(94,226)(95,227)(96,228)(97,229)(98,230)(99,231)(100,232)(101,233)(102,234)(103,235)(104,236)(105,237)(106,238)(107,239)(108,240)(109,241)(110,242)(111,243)(112,244)(113,245)(114,246)(115,247)(116,248)(117,249)(118,250)(119,251)(120,252)(121,253)(122,254)(123,255)(124,256)(125,257)(126,258)(127,259)(128,260)(129,261)(130,262)(131,263)(132,264)(133,265)(134,266)(135,267)(136,268)(137,269)(138,270)(139,271)(140,272), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,225)(155,226)(156,227)(157,228)(158,229)(159,230)(160,231)(161,232)(162,233)(163,234)(164,235)(165,236)(166,237)(167,238)(168,239)(169,240)(170,241)(171,242)(172,243)(173,244)(174,245)(175,246)(176,247)(177,248)(178,249)(179,250)(180,251)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257)(187,258)(188,259)(189,260)(190,261)(191,262)(192,263)(193,264)(194,265)(195,266)(196,267)(197,268)(198,269)(199,270)(200,271)(201,272)(202,273)(203,274)(204,275)(205,276)(206,277)(207,278)(208,279)(209,280)(210,211), (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([[(1,192),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,199),(9,200),(10,201),(11,202),(12,203),(13,204),(14,205),(15,206),(16,207),(17,208),(18,209),(19,210),(20,141),(21,142),(22,143),(23,144),(24,145),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,153),(33,154),(34,155),(35,156),(36,157),(37,158),(38,159),(39,160),(40,161),(41,162),(42,163),(43,164),(44,165),(45,166),(46,167),(47,168),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,178),(58,179),(59,180),(60,181),(61,182),(62,183),(63,184),(64,185),(65,186),(66,187),(67,188),(68,189),(69,190),(70,191),(71,273),(72,274),(73,275),(74,276),(75,277),(76,278),(77,279),(78,280),(79,211),(80,212),(81,213),(82,214),(83,215),(84,216),(85,217),(86,218),(87,219),(88,220),(89,221),(90,222),(91,223),(92,224),(93,225),(94,226),(95,227),(96,228),(97,229),(98,230),(99,231),(100,232),(101,233),(102,234),(103,235),(104,236),(105,237),(106,238),(107,239),(108,240),(109,241),(110,242),(111,243),(112,244),(113,245),(114,246),(115,247),(116,248),(117,249),(118,250),(119,251),(120,252),(121,253),(122,254),(123,255),(124,256),(125,257),(126,258),(127,259),(128,260),(129,261),(130,262),(131,263),(132,264),(133,265),(134,266),(135,267),(136,268),(137,269),(138,270),(139,271),(140,272)], [(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,218),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,225),(155,226),(156,227),(157,228),(158,229),(159,230),(160,231),(161,232),(162,233),(163,234),(164,235),(165,236),(166,237),(167,238),(168,239),(169,240),(170,241),(171,242),(172,243),(173,244),(174,245),(175,246),(176,247),(177,248),(178,249),(179,250),(180,251),(181,252),(182,253),(183,254),(184,255),(185,256),(186,257),(187,258),(188,259),(189,260),(190,261),(191,262),(192,263),(193,264),(194,265),(195,266),(196,267),(197,268),(198,269),(199,270),(200,271),(201,272),(202,273),(203,274),(204,275),(205,276),(206,277),(207,278),(208,279),(209,280),(210,211)], [(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 | ··· | 2G | 5A | 5B | 5C | 5D | 7A | ··· | 7F | 10A | ··· | 10AB | 14A | ··· | 14AP | 35A | ··· | 35X | 70A | ··· | 70FL |
| order | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 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 |
| type | + | + | ||||||
| image | C1 | C2 | C5 | C7 | C10 | C14 | C35 | C70 |
| kernel | C22×C70 | C2×C70 | C22×C14 | C22×C10 | C2×C14 | C2×C10 | C23 | C22 |
| # reps | 1 | 7 | 4 | 6 | 28 | 42 | 24 | 168 |
Matrix representation of C22×C70 ►in GL3(𝔽71) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 70 |
| 70 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 23 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 63 |
G:=sub<GL(3,GF(71))| [1,0,0,0,1,0,0,0,70],[70,0,0,0,1,0,0,0,1],[23,0,0,0,23,0,0,0,63] >;
C22×C70 in GAP, Magma, Sage, TeX
C_2^2\times C_{70} % in TeX
G:=Group("C2^2xC70"); // GroupNames label
G:=SmallGroup(280,40);
// by ID
G=gap.SmallGroup(280,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^70=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations