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G = C22×C80order 320 = 26·5

Abelian group of type [2,2,80]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C80, SmallGroup(320,1003)

Series: Derived Chief Lower central Upper central

C1 — C22×C80
C1C2C4C8C40C80C2×C80 — C22×C80
C1 — C22×C80
C1 — C22×C80

Generators and relations for C22×C80
 G = < a,b,c | a2=b2=c80=1, ab=ba, ac=ca, bc=cb >

Subgroups: 98, all normal (16 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×7], C5, C8, C8 [×3], C2×C4 [×6], C23, C10, C10 [×6], C16 [×4], C2×C8 [×6], C22×C4, C20, C20 [×3], C2×C10 [×7], C2×C16 [×6], C22×C8, C40, C40 [×3], C2×C20 [×6], C22×C10, C22×C16, C80 [×4], C2×C40 [×6], C22×C20, C2×C80 [×6], C22×C40, C22×C80
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10 [×7], C16 [×4], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C2×C16 [×6], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C22×C16, C80 [×4], C2×C40 [×6], C22×C20, C2×C80 [×6], C22×C40, C22×C80

Smallest permutation representation of C22×C80
Regular action on 320 points
Generators in S320
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(161 252)(162 253)(163 254)(164 255)(165 256)(166 257)(167 258)(168 259)(169 260)(170 261)(171 262)(172 263)(173 264)(174 265)(175 266)(176 267)(177 268)(178 269)(179 270)(180 271)(181 272)(182 273)(183 274)(184 275)(185 276)(186 277)(187 278)(188 279)(189 280)(190 281)(191 282)(192 283)(193 284)(194 285)(195 286)(196 287)(197 288)(198 289)(199 290)(200 291)(201 292)(202 293)(203 294)(204 295)(205 296)(206 297)(207 298)(208 299)(209 300)(210 301)(211 302)(212 303)(213 304)(214 305)(215 306)(216 307)(217 308)(218 309)(219 310)(220 311)(221 312)(222 313)(223 314)(224 315)(225 316)(226 317)(227 318)(228 319)(229 320)(230 241)(231 242)(232 243)(233 244)(234 245)(235 246)(236 247)(237 248)(238 249)(239 250)(240 251)
(1 167)(2 168)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 175)(10 176)(11 177)(12 178)(13 179)(14 180)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 209)(44 210)(45 211)(46 212)(47 213)(48 214)(49 215)(50 216)(51 217)(52 218)(53 219)(54 220)(55 221)(56 222)(57 223)(58 224)(59 225)(60 226)(61 227)(62 228)(63 229)(64 230)(65 231)(66 232)(67 233)(68 234)(69 235)(70 236)(71 237)(72 238)(73 239)(74 240)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 275)(82 276)(83 277)(84 278)(85 279)(86 280)(87 281)(88 282)(89 283)(90 284)(91 285)(92 286)(93 287)(94 288)(95 289)(96 290)(97 291)(98 292)(99 293)(100 294)(101 295)(102 296)(103 297)(104 298)(105 299)(106 300)(107 301)(108 302)(109 303)(110 304)(111 305)(112 306)(113 307)(114 308)(115 309)(116 310)(117 311)(118 312)(119 313)(120 314)(121 315)(122 316)(123 317)(124 318)(125 319)(126 320)(127 241)(128 242)(129 243)(130 244)(131 245)(132 246)(133 247)(134 248)(135 249)(136 250)(137 251)(138 252)(139 253)(140 254)(141 255)(142 256)(143 257)(144 258)(145 259)(146 260)(147 261)(148 262)(149 263)(150 264)(151 265)(152 266)(153 267)(154 268)(155 269)(156 270)(157 271)(158 272)(159 273)(160 274)
(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)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320)

G:=sub<Sym(320)| (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(161,252)(162,253)(163,254)(164,255)(165,256)(166,257)(167,258)(168,259)(169,260)(170,261)(171,262)(172,263)(173,264)(174,265)(175,266)(176,267)(177,268)(178,269)(179,270)(180,271)(181,272)(182,273)(183,274)(184,275)(185,276)(186,277)(187,278)(188,279)(189,280)(190,281)(191,282)(192,283)(193,284)(194,285)(195,286)(196,287)(197,288)(198,289)(199,290)(200,291)(201,292)(202,293)(203,294)(204,295)(205,296)(206,297)(207,298)(208,299)(209,300)(210,301)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,316)(226,317)(227,318)(228,319)(229,320)(230,241)(231,242)(232,243)(233,244)(234,245)(235,246)(236,247)(237,248)(238,249)(239,250)(240,251), (1,167)(2,168)(3,169)(4,170)(5,171)(6,172)(7,173)(8,174)(9,175)(10,176)(11,177)(12,178)(13,179)(14,180)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,227)(62,228)(63,229)(64,230)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(71,237)(72,238)(73,239)(74,240)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,275)(82,276)(83,277)(84,278)(85,279)(86,280)(87,281)(88,282)(89,283)(90,284)(91,285)(92,286)(93,287)(94,288)(95,289)(96,290)(97,291)(98,292)(99,293)(100,294)(101,295)(102,296)(103,297)(104,298)(105,299)(106,300)(107,301)(108,302)(109,303)(110,304)(111,305)(112,306)(113,307)(114,308)(115,309)(116,310)(117,311)(118,312)(119,313)(120,314)(121,315)(122,316)(123,317)(124,318)(125,319)(126,320)(127,241)(128,242)(129,243)(130,244)(131,245)(132,246)(133,247)(134,248)(135,249)(136,250)(137,251)(138,252)(139,253)(140,254)(141,255)(142,256)(143,257)(144,258)(145,259)(146,260)(147,261)(148,262)(149,263)(150,264)(151,265)(152,266)(153,267)(154,268)(155,269)(156,270)(157,271)(158,272)(159,273)(160,274), (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)>;

G:=Group( (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,128)(66,129)(67,130)(68,131)(69,132)(70,133)(71,134)(72,135)(73,136)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(161,252)(162,253)(163,254)(164,255)(165,256)(166,257)(167,258)(168,259)(169,260)(170,261)(171,262)(172,263)(173,264)(174,265)(175,266)(176,267)(177,268)(178,269)(179,270)(180,271)(181,272)(182,273)(183,274)(184,275)(185,276)(186,277)(187,278)(188,279)(189,280)(190,281)(191,282)(192,283)(193,284)(194,285)(195,286)(196,287)(197,288)(198,289)(199,290)(200,291)(201,292)(202,293)(203,294)(204,295)(205,296)(206,297)(207,298)(208,299)(209,300)(210,301)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)(224,315)(225,316)(226,317)(227,318)(228,319)(229,320)(230,241)(231,242)(232,243)(233,244)(234,245)(235,246)(236,247)(237,248)(238,249)(239,250)(240,251), (1,167)(2,168)(3,169)(4,170)(5,171)(6,172)(7,173)(8,174)(9,175)(10,176)(11,177)(12,178)(13,179)(14,180)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,223)(58,224)(59,225)(60,226)(61,227)(62,228)(63,229)(64,230)(65,231)(66,232)(67,233)(68,234)(69,235)(70,236)(71,237)(72,238)(73,239)(74,240)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,275)(82,276)(83,277)(84,278)(85,279)(86,280)(87,281)(88,282)(89,283)(90,284)(91,285)(92,286)(93,287)(94,288)(95,289)(96,290)(97,291)(98,292)(99,293)(100,294)(101,295)(102,296)(103,297)(104,298)(105,299)(106,300)(107,301)(108,302)(109,303)(110,304)(111,305)(112,306)(113,307)(114,308)(115,309)(116,310)(117,311)(118,312)(119,313)(120,314)(121,315)(122,316)(123,317)(124,318)(125,319)(126,320)(127,241)(128,242)(129,243)(130,244)(131,245)(132,246)(133,247)(134,248)(135,249)(136,250)(137,251)(138,252)(139,253)(140,254)(141,255)(142,256)(143,257)(144,258)(145,259)(146,260)(147,261)(148,262)(149,263)(150,264)(151,265)(152,266)(153,267)(154,268)(155,269)(156,270)(157,271)(158,272)(159,273)(160,274), (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320) );

G=PermutationGroup([(1,144),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,160),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,128),(66,129),(67,130),(68,131),(69,132),(70,133),(71,134),(72,135),(73,136),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(161,252),(162,253),(163,254),(164,255),(165,256),(166,257),(167,258),(168,259),(169,260),(170,261),(171,262),(172,263),(173,264),(174,265),(175,266),(176,267),(177,268),(178,269),(179,270),(180,271),(181,272),(182,273),(183,274),(184,275),(185,276),(186,277),(187,278),(188,279),(189,280),(190,281),(191,282),(192,283),(193,284),(194,285),(195,286),(196,287),(197,288),(198,289),(199,290),(200,291),(201,292),(202,293),(203,294),(204,295),(205,296),(206,297),(207,298),(208,299),(209,300),(210,301),(211,302),(212,303),(213,304),(214,305),(215,306),(216,307),(217,308),(218,309),(219,310),(220,311),(221,312),(222,313),(223,314),(224,315),(225,316),(226,317),(227,318),(228,319),(229,320),(230,241),(231,242),(232,243),(233,244),(234,245),(235,246),(236,247),(237,248),(238,249),(239,250),(240,251)], [(1,167),(2,168),(3,169),(4,170),(5,171),(6,172),(7,173),(8,174),(9,175),(10,176),(11,177),(12,178),(13,179),(14,180),(15,181),(16,182),(17,183),(18,184),(19,185),(20,186),(21,187),(22,188),(23,189),(24,190),(25,191),(26,192),(27,193),(28,194),(29,195),(30,196),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,209),(44,210),(45,211),(46,212),(47,213),(48,214),(49,215),(50,216),(51,217),(52,218),(53,219),(54,220),(55,221),(56,222),(57,223),(58,224),(59,225),(60,226),(61,227),(62,228),(63,229),(64,230),(65,231),(66,232),(67,233),(68,234),(69,235),(70,236),(71,237),(72,238),(73,239),(74,240),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,275),(82,276),(83,277),(84,278),(85,279),(86,280),(87,281),(88,282),(89,283),(90,284),(91,285),(92,286),(93,287),(94,288),(95,289),(96,290),(97,291),(98,292),(99,293),(100,294),(101,295),(102,296),(103,297),(104,298),(105,299),(106,300),(107,301),(108,302),(109,303),(110,304),(111,305),(112,306),(113,307),(114,308),(115,309),(116,310),(117,311),(118,312),(119,313),(120,314),(121,315),(122,316),(123,317),(124,318),(125,319),(126,320),(127,241),(128,242),(129,243),(130,244),(131,245),(132,246),(133,247),(134,248),(135,249),(136,250),(137,251),(138,252),(139,253),(140,254),(141,255),(142,256),(143,257),(144,258),(145,259),(146,260),(147,261),(148,262),(149,263),(150,264),(151,265),(152,266),(153,267),(154,268),(155,269),(156,270),(157,271),(158,272),(159,273),(160,274)], [(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),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320)])

320 conjugacy classes

class 1 2A···2G4A···4H5A5B5C5D8A···8P10A···10AB16A···16AF20A···20AF40A···40BL80A···80DX
order12···24···455558···810···1016···1620···2040···4080···80
size11···11···111111···11···11···11···11···11···1

320 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C4C5C8C8C10C10C16C20C20C40C40C80
kernelC22×C80C2×C80C22×C40C2×C40C22×C20C22×C16C2×C20C22×C10C2×C16C22×C8C2×C10C2×C8C22×C4C2×C4C23C22
# reps161624124244322484816128

Matrix representation of C22×C80 in GL3(𝔽241) generated by

100
02400
001
,
100
02400
00240
,
11100
01360
0073
G:=sub<GL(3,GF(241))| [1,0,0,0,240,0,0,0,1],[1,0,0,0,240,0,0,0,240],[111,0,0,0,136,0,0,0,73] >;

C22×C80 in GAP, Magma, Sage, TeX

C_2^2\times C_{80}
% in TeX

G:=Group("C2^2xC80");
// GroupNames label

G:=SmallGroup(320,1003);
// by ID

G=gap.SmallGroup(320,1003);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^80=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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