direct product, abelian, monomial, 2-elementary
Aliases: C2×C4×C40, SmallGroup(320,903)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C4×C40 |
| C1 — C2×C4×C40 |
| C1 — C2×C4×C40 |
Generators and relations for C2×C4×C40
G = < a,b,c | a2=b4=c40=1, ab=ba, ac=ca, bc=cb >
Subgroups: 162, all normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, C20, C2×C10, C2×C10, C4×C8, C2×C42, C22×C8, C40, C2×C20, C2×C20, C22×C10, C2×C4×C8, C4×C20, C2×C40, C22×C20, C22×C20, C4×C40, C2×C4×C20, C22×C40, C2×C4×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C42, C2×C8, C22×C4, C20, C2×C10, C4×C8, C2×C42, C22×C8, C40, C2×C20, C22×C10, C2×C4×C8, C4×C20, C2×C40, C22×C20, C4×C40, C2×C4×C20, C22×C40, C2×C4×C40
►Regular action on 320 points
Generators in S320
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 81)(40 82)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(161 261)(162 262)(163 263)(164 264)(165 265)(166 266)(167 267)(168 268)(169 269)(170 270)(171 271)(172 272)(173 273)(174 274)(175 275)(176 276)(177 277)(178 278)(179 279)(180 280)(181 241)(182 242)(183 243)(184 244)(185 245)(186 246)(187 247)(188 248)(189 249)(190 250)(191 251)(192 252)(193 253)(194 254)(195 255)(196 256)(197 257)(198 258)(199 259)(200 260)(201 286)(202 287)(203 288)(204 289)(205 290)(206 291)(207 292)(208 293)(209 294)(210 295)(211 296)(212 297)(213 298)(214 299)(215 300)(216 301)(217 302)(218 303)(219 304)(220 305)(221 306)(222 307)(223 308)(224 309)(225 310)(226 311)(227 312)(228 313)(229 314)(230 315)(231 316)(232 317)(233 318)(234 319)(235 320)(236 281)(237 282)(238 283)(239 284)(240 285)
(1 218 63 183)(2 219 64 184)(3 220 65 185)(4 221 66 186)(5 222 67 187)(6 223 68 188)(7 224 69 189)(8 225 70 190)(9 226 71 191)(10 227 72 192)(11 228 73 193)(12 229 74 194)(13 230 75 195)(14 231 76 196)(15 232 77 197)(16 233 78 198)(17 234 79 199)(18 235 80 200)(19 236 41 161)(20 237 42 162)(21 238 43 163)(22 239 44 164)(23 240 45 165)(24 201 46 166)(25 202 47 167)(26 203 48 168)(27 204 49 169)(28 205 50 170)(29 206 51 171)(30 207 52 172)(31 208 53 173)(32 209 54 174)(33 210 55 175)(34 211 56 176)(35 212 57 177)(36 213 58 178)(37 214 59 179)(38 215 60 180)(39 216 61 181)(40 217 62 182)(81 301 149 241)(82 302 150 242)(83 303 151 243)(84 304 152 244)(85 305 153 245)(86 306 154 246)(87 307 155 247)(88 308 156 248)(89 309 157 249)(90 310 158 250)(91 311 159 251)(92 312 160 252)(93 313 121 253)(94 314 122 254)(95 315 123 255)(96 316 124 256)(97 317 125 257)(98 318 126 258)(99 319 127 259)(100 320 128 260)(101 281 129 261)(102 282 130 262)(103 283 131 263)(104 284 132 264)(105 285 133 265)(106 286 134 266)(107 287 135 267)(108 288 136 268)(109 289 137 269)(110 290 138 270)(111 291 139 271)(112 292 140 272)(113 293 141 273)(114 294 142 274)(115 295 143 275)(116 296 144 276)(117 297 145 277)(118 298 146 278)(119 299 147 279)(120 300 148 280)
(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)| (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,81)(40,82)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(161,261)(162,262)(163,263)(164,264)(165,265)(166,266)(167,267)(168,268)(169,269)(170,270)(171,271)(172,272)(173,273)(174,274)(175,275)(176,276)(177,277)(178,278)(179,279)(180,280)(181,241)(182,242)(183,243)(184,244)(185,245)(186,246)(187,247)(188,248)(189,249)(190,250)(191,251)(192,252)(193,253)(194,254)(195,255)(196,256)(197,257)(198,258)(199,259)(200,260)(201,286)(202,287)(203,288)(204,289)(205,290)(206,291)(207,292)(208,293)(209,294)(210,295)(211,296)(212,297)(213,298)(214,299)(215,300)(216,301)(217,302)(218,303)(219,304)(220,305)(221,306)(222,307)(223,308)(224,309)(225,310)(226,311)(227,312)(228,313)(229,314)(230,315)(231,316)(232,317)(233,318)(234,319)(235,320)(236,281)(237,282)(238,283)(239,284)(240,285), (1,218,63,183)(2,219,64,184)(3,220,65,185)(4,221,66,186)(5,222,67,187)(6,223,68,188)(7,224,69,189)(8,225,70,190)(9,226,71,191)(10,227,72,192)(11,228,73,193)(12,229,74,194)(13,230,75,195)(14,231,76,196)(15,232,77,197)(16,233,78,198)(17,234,79,199)(18,235,80,200)(19,236,41,161)(20,237,42,162)(21,238,43,163)(22,239,44,164)(23,240,45,165)(24,201,46,166)(25,202,47,167)(26,203,48,168)(27,204,49,169)(28,205,50,170)(29,206,51,171)(30,207,52,172)(31,208,53,173)(32,209,54,174)(33,210,55,175)(34,211,56,176)(35,212,57,177)(36,213,58,178)(37,214,59,179)(38,215,60,180)(39,216,61,181)(40,217,62,182)(81,301,149,241)(82,302,150,242)(83,303,151,243)(84,304,152,244)(85,305,153,245)(86,306,154,246)(87,307,155,247)(88,308,156,248)(89,309,157,249)(90,310,158,250)(91,311,159,251)(92,312,160,252)(93,313,121,253)(94,314,122,254)(95,315,123,255)(96,316,124,256)(97,317,125,257)(98,318,126,258)(99,319,127,259)(100,320,128,260)(101,281,129,261)(102,282,130,262)(103,283,131,263)(104,284,132,264)(105,285,133,265)(106,286,134,266)(107,287,135,267)(108,288,136,268)(109,289,137,269)(110,290,138,270)(111,291,139,271)(112,292,140,272)(113,293,141,273)(114,294,142,274)(115,295,143,275)(116,296,144,276)(117,297,145,277)(118,298,146,278)(119,299,147,279)(120,300,148,280), (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( (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,101)(20,102)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,81)(40,82)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(161,261)(162,262)(163,263)(164,264)(165,265)(166,266)(167,267)(168,268)(169,269)(170,270)(171,271)(172,272)(173,273)(174,274)(175,275)(176,276)(177,277)(178,278)(179,279)(180,280)(181,241)(182,242)(183,243)(184,244)(185,245)(186,246)(187,247)(188,248)(189,249)(190,250)(191,251)(192,252)(193,253)(194,254)(195,255)(196,256)(197,257)(198,258)(199,259)(200,260)(201,286)(202,287)(203,288)(204,289)(205,290)(206,291)(207,292)(208,293)(209,294)(210,295)(211,296)(212,297)(213,298)(214,299)(215,300)(216,301)(217,302)(218,303)(219,304)(220,305)(221,306)(222,307)(223,308)(224,309)(225,310)(226,311)(227,312)(228,313)(229,314)(230,315)(231,316)(232,317)(233,318)(234,319)(235,320)(236,281)(237,282)(238,283)(239,284)(240,285), (1,218,63,183)(2,219,64,184)(3,220,65,185)(4,221,66,186)(5,222,67,187)(6,223,68,188)(7,224,69,189)(8,225,70,190)(9,226,71,191)(10,227,72,192)(11,228,73,193)(12,229,74,194)(13,230,75,195)(14,231,76,196)(15,232,77,197)(16,233,78,198)(17,234,79,199)(18,235,80,200)(19,236,41,161)(20,237,42,162)(21,238,43,163)(22,239,44,164)(23,240,45,165)(24,201,46,166)(25,202,47,167)(26,203,48,168)(27,204,49,169)(28,205,50,170)(29,206,51,171)(30,207,52,172)(31,208,53,173)(32,209,54,174)(33,210,55,175)(34,211,56,176)(35,212,57,177)(36,213,58,178)(37,214,59,179)(38,215,60,180)(39,216,61,181)(40,217,62,182)(81,301,149,241)(82,302,150,242)(83,303,151,243)(84,304,152,244)(85,305,153,245)(86,306,154,246)(87,307,155,247)(88,308,156,248)(89,309,157,249)(90,310,158,250)(91,311,159,251)(92,312,160,252)(93,313,121,253)(94,314,122,254)(95,315,123,255)(96,316,124,256)(97,317,125,257)(98,318,126,258)(99,319,127,259)(100,320,128,260)(101,281,129,261)(102,282,130,262)(103,283,131,263)(104,284,132,264)(105,285,133,265)(106,286,134,266)(107,287,135,267)(108,288,136,268)(109,289,137,269)(110,290,138,270)(111,291,139,271)(112,292,140,272)(113,293,141,273)(114,294,142,274)(115,295,143,275)(116,296,144,276)(117,297,145,277)(118,298,146,278)(119,299,147,279)(120,300,148,280), (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([[(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,101),(20,102),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,81),(40,82),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(161,261),(162,262),(163,263),(164,264),(165,265),(166,266),(167,267),(168,268),(169,269),(170,270),(171,271),(172,272),(173,273),(174,274),(175,275),(176,276),(177,277),(178,278),(179,279),(180,280),(181,241),(182,242),(183,243),(184,244),(185,245),(186,246),(187,247),(188,248),(189,249),(190,250),(191,251),(192,252),(193,253),(194,254),(195,255),(196,256),(197,257),(198,258),(199,259),(200,260),(201,286),(202,287),(203,288),(204,289),(205,290),(206,291),(207,292),(208,293),(209,294),(210,295),(211,296),(212,297),(213,298),(214,299),(215,300),(216,301),(217,302),(218,303),(219,304),(220,305),(221,306),(222,307),(223,308),(224,309),(225,310),(226,311),(227,312),(228,313),(229,314),(230,315),(231,316),(232,317),(233,318),(234,319),(235,320),(236,281),(237,282),(238,283),(239,284),(240,285)], [(1,218,63,183),(2,219,64,184),(3,220,65,185),(4,221,66,186),(5,222,67,187),(6,223,68,188),(7,224,69,189),(8,225,70,190),(9,226,71,191),(10,227,72,192),(11,228,73,193),(12,229,74,194),(13,230,75,195),(14,231,76,196),(15,232,77,197),(16,233,78,198),(17,234,79,199),(18,235,80,200),(19,236,41,161),(20,237,42,162),(21,238,43,163),(22,239,44,164),(23,240,45,165),(24,201,46,166),(25,202,47,167),(26,203,48,168),(27,204,49,169),(28,205,50,170),(29,206,51,171),(30,207,52,172),(31,208,53,173),(32,209,54,174),(33,210,55,175),(34,211,56,176),(35,212,57,177),(36,213,58,178),(37,214,59,179),(38,215,60,180),(39,216,61,181),(40,217,62,182),(81,301,149,241),(82,302,150,242),(83,303,151,243),(84,304,152,244),(85,305,153,245),(86,306,154,246),(87,307,155,247),(88,308,156,248),(89,309,157,249),(90,310,158,250),(91,311,159,251),(92,312,160,252),(93,313,121,253),(94,314,122,254),(95,315,123,255),(96,316,124,256),(97,317,125,257),(98,318,126,258),(99,319,127,259),(100,320,128,260),(101,281,129,261),(102,282,130,262),(103,283,131,263),(104,284,132,264),(105,285,133,265),(106,286,134,266),(107,287,135,267),(108,288,136,268),(109,289,137,269),(110,290,138,270),(111,291,139,271),(112,292,140,272),(113,293,141,273),(114,294,142,274),(115,295,143,275),(116,296,144,276),(117,297,145,277),(118,298,146,278),(119,299,147,279),(120,300,148,280)], [(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 | ··· | 2G | 4A | ··· | 4X | 5A | 5B | 5C | 5D | 8A | ··· | 8AF | 10A | ··· | 10AB | 20A | ··· | 20CR | 40A | ··· | 40DX |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
320 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C20 | C20 | C20 | C40 |
| kernel | C2×C4×C40 | C4×C40 | C2×C4×C20 | C22×C40 | C4×C20 | C2×C40 | C22×C20 | C2×C4×C8 | C2×C20 | C4×C8 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 4 | 32 | 16 | 4 | 8 | 16 | 64 | 16 | 128 |
Matrix representation of C2×C4×C40 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 9 |
| 3 | 0 | 0 |
| 0 | 20 | 0 |
| 0 | 0 | 21 |
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,1],[1,0,0,0,1,0,0,0,9],[3,0,0,0,20,0,0,0,21] >;
C2×C4×C40 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{40} % in TeX
G:=Group("C2xC4xC40"); // GroupNames label
G:=SmallGroup(320,903);
// by ID
G=gap.SmallGroup(320,903);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,568,172]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^40=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations