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## G = C22×C70order 280 = 23·5·7

### Abelian group of type [2,2,70]

Aliases: C22×C70, SmallGroup(280,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C70
 Chief series C1 — C7 — C35 — C70 — C2×C70 — C22×C70
 Lower central C1 — C22×C70
 Upper central 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 [×7], C22 [×7], C5, C7, C23, C10 [×7], C14 [×7], C2×C10 [×7], C2×C14 [×7], C35, C22×C10, C22×C14, C70 [×7], C2×C70 [×7], C22×C70
Quotients: C1, C2 [×7], C22 [×7], C5, C7, C23, C10 [×7], C14 [×7], C2×C10 [×7], C2×C14 [×7], C35, C22×C10, C22×C14, C70 [×7], C2×C70 [×7], C22×C70

Smallest permutation representation of C22×C70
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)(54 141)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 238)(72 239)(73 240)(74 241)(75 242)(76 243)(77 244)(78 245)(79 246)(80 247)(81 248)(82 249)(83 250)(84 251)(85 252)(86 253)(87 254)(88 255)(89 256)(90 257)(91 258)(92 259)(93 260)(94 261)(95 262)(96 263)(97 264)(98 265)(99 266)(100 267)(101 268)(102 269)(103 270)(104 271)(105 272)(106 273)(107 274)(108 275)(109 276)(110 277)(111 278)(112 279)(113 280)(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)(139 236)(140 237)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 80)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(141 237)(142 238)(143 239)(144 240)(145 241)(146 242)(147 243)(148 244)(149 245)(150 246)(151 247)(152 248)(153 249)(154 250)(155 251)(156 252)(157 253)(158 254)(159 255)(160 256)(161 257)(162 258)(163 259)(164 260)(165 261)(166 262)(167 263)(168 264)(169 265)(170 266)(171 267)(172 268)(173 269)(174 270)(175 271)(176 272)(177 273)(178 274)(179 275)(180 276)(181 277)(182 278)(183 279)(184 280)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)(197 223)(198 224)(199 225)(200 226)(201 227)(202 228)(203 229)(204 230)(205 231)(206 232)(207 233)(208 234)(209 235)(210 236)
(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)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,238)(72,239)(73,240)(74,241)(75,242)(76,243)(77,244)(78,245)(79,246)(80,247)(81,248)(82,249)(83,250)(84,251)(85,252)(86,253)(87,254)(88,255)(89,256)(90,257)(91,258)(92,259)(93,260)(94,261)(95,262)(96,263)(97,264)(98,265)(99,266)(100,267)(101,268)(102,269)(103,270)(104,271)(105,272)(106,273)(107,274)(108,275)(109,276)(110,277)(111,278)(112,279)(113,280)(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)(139,236)(140,237), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(141,237)(142,238)(143,239)(144,240)(145,241)(146,242)(147,243)(148,244)(149,245)(150,246)(151,247)(152,248)(153,249)(154,250)(155,251)(156,252)(157,253)(158,254)(159,255)(160,256)(161,257)(162,258)(163,259)(164,260)(165,261)(166,262)(167,263)(168,264)(169,265)(170,266)(171,267)(172,268)(173,269)(174,270)(175,271)(176,272)(177,273)(178,274)(179,275)(180,276)(181,277)(182,278)(183,279)(184,280)(185,211)(186,212)(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221)(196,222)(197,223)(198,224)(199,225)(200,226)(201,227)(202,228)(203,229)(204,230)(205,231)(206,232)(207,233)(208,234)(209,235)(210,236), (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)(54,141)(55,142)(56,143)(57,144)(58,145)(59,146)(60,147)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,238)(72,239)(73,240)(74,241)(75,242)(76,243)(77,244)(78,245)(79,246)(80,247)(81,248)(82,249)(83,250)(84,251)(85,252)(86,253)(87,254)(88,255)(89,256)(90,257)(91,258)(92,259)(93,260)(94,261)(95,262)(96,263)(97,264)(98,265)(99,266)(100,267)(101,268)(102,269)(103,270)(104,271)(105,272)(106,273)(107,274)(108,275)(109,276)(110,277)(111,278)(112,279)(113,280)(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)(139,236)(140,237), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79)(64,80)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(141,237)(142,238)(143,239)(144,240)(145,241)(146,242)(147,243)(148,244)(149,245)(150,246)(151,247)(152,248)(153,249)(154,250)(155,251)(156,252)(157,253)(158,254)(159,255)(160,256)(161,257)(162,258)(163,259)(164,260)(165,261)(166,262)(167,263)(168,264)(169,265)(170,266)(171,267)(172,268)(173,269)(174,270)(175,271)(176,272)(177,273)(178,274)(179,275)(180,276)(181,277)(182,278)(183,279)(184,280)(185,211)(186,212)(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221)(196,222)(197,223)(198,224)(199,225)(200,226)(201,227)(202,228)(203,229)(204,230)(205,231)(206,232)(207,233)(208,234)(209,235)(210,236), (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),(54,141),(55,142),(56,143),(57,144),(58,145),(59,146),(60,147),(61,148),(62,149),(63,150),(64,151),(65,152),(66,153),(67,154),(68,155),(69,156),(70,157),(71,238),(72,239),(73,240),(74,241),(75,242),(76,243),(77,244),(78,245),(79,246),(80,247),(81,248),(82,249),(83,250),(84,251),(85,252),(86,253),(87,254),(88,255),(89,256),(90,257),(91,258),(92,259),(93,260),(94,261),(95,262),(96,263),(97,264),(98,265),(99,266),(100,267),(101,268),(102,269),(103,270),(104,271),(105,272),(106,273),(107,274),(108,275),(109,276),(110,277),(111,278),(112,279),(113,280),(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),(139,236),(140,237)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79),(64,80),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(141,237),(142,238),(143,239),(144,240),(145,241),(146,242),(147,243),(148,244),(149,245),(150,246),(151,247),(152,248),(153,249),(154,250),(155,251),(156,252),(157,253),(158,254),(159,255),(160,256),(161,257),(162,258),(163,259),(164,260),(165,261),(166,262),(167,263),(168,264),(169,265),(170,266),(171,267),(172,268),(173,269),(174,270),(175,271),(176,272),(177,273),(178,274),(179,275),(180,276),(181,277),(182,278),(183,279),(184,280),(185,211),(186,212),(187,213),(188,214),(189,215),(190,216),(191,217),(192,218),(193,219),(194,220),(195,221),(196,222),(197,223),(198,224),(199,225),(200,226),(201,227),(202,228),(203,229),(204,230),(205,231),(206,232),(207,233),(208,234),(209,235),(210,236)], [(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

׿
×
𝔽