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## G = C2×C4×C40order 320 = 26·5

### Abelian group of type [2,4,40]

Aliases: C2×C4×C40, SmallGroup(320,903)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C40
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C40 — C4×C40 — C2×C4×C40
 Lower central C1 — C2×C4×C40
 Upper central 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 [×6], C4 [×12], C22, C22 [×6], C5, C8 [×8], C2×C4 [×2], C2×C4 [×16], C23, C10, C10 [×6], C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], C20 [×12], C2×C10, C2×C10 [×6], C4×C8 [×4], C2×C42, C22×C8 [×2], C40 [×8], C2×C20 [×2], C2×C20 [×16], C22×C10, C2×C4×C8, C4×C20 [×4], C2×C40 [×12], C22×C20, C22×C20 [×2], C4×C40 [×4], C2×C4×C20, C22×C40 [×2], C2×C4×C40
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C5, C8 [×8], C2×C4 [×18], C23, C10 [×7], C42 [×4], C2×C8 [×12], C22×C4 [×3], C20 [×12], C2×C10 [×7], C4×C8 [×4], C2×C42, C22×C8 [×2], C40 [×8], C2×C20 [×18], C22×C10, C2×C4×C8, C4×C20 [×4], C2×C40 [×12], C22×C20 [×3], C4×C40 [×4], C2×C4×C20, C22×C40 [×2], C2×C4×C40

Smallest permutation representation of C2×C4×C40
Regular action on 320 points
Generators in S320
(1 308)(2 309)(3 310)(4 311)(5 312)(6 313)(7 314)(8 315)(9 316)(10 317)(11 318)(12 319)(13 320)(14 281)(15 282)(16 283)(17 284)(18 285)(19 286)(20 287)(21 288)(22 289)(23 290)(24 291)(25 292)(26 293)(27 294)(28 295)(29 296)(30 297)(31 298)(32 299)(33 300)(34 301)(35 302)(36 303)(37 304)(38 305)(39 306)(40 307)(41 255)(42 256)(43 257)(44 258)(45 259)(46 260)(47 261)(48 262)(49 263)(50 264)(51 265)(52 266)(53 267)(54 268)(55 269)(56 270)(57 271)(58 272)(59 273)(60 274)(61 275)(62 276)(63 277)(64 278)(65 279)(66 280)(67 241)(68 242)(69 243)(70 244)(71 245)(72 246)(73 247)(74 248)(75 249)(76 250)(77 251)(78 252)(79 253)(80 254)(81 181)(82 182)(83 183)(84 184)(85 185)(86 186)(87 187)(88 188)(89 189)(90 190)(91 191)(92 192)(93 193)(94 194)(95 195)(96 196)(97 197)(98 198)(99 199)(100 200)(101 161)(102 162)(103 163)(104 164)(105 165)(106 166)(107 167)(108 168)(109 169)(110 170)(111 171)(112 172)(113 173)(114 174)(115 175)(116 176)(117 177)(118 178)(119 179)(120 180)(121 216)(122 217)(123 218)(124 219)(125 220)(126 221)(127 222)(128 223)(129 224)(130 225)(131 226)(132 227)(133 228)(134 229)(135 230)(136 231)(137 232)(138 233)(139 234)(140 235)(141 236)(142 237)(143 238)(144 239)(145 240)(146 201)(147 202)(148 203)(149 204)(150 205)(151 206)(152 207)(153 208)(154 209)(155 210)(156 211)(157 212)(158 213)(159 214)(160 215)
(1 245 160 99)(2 246 121 100)(3 247 122 101)(4 248 123 102)(5 249 124 103)(6 250 125 104)(7 251 126 105)(8 252 127 106)(9 253 128 107)(10 254 129 108)(11 255 130 109)(12 256 131 110)(13 257 132 111)(14 258 133 112)(15 259 134 113)(16 260 135 114)(17 261 136 115)(18 262 137 116)(19 263 138 117)(20 264 139 118)(21 265 140 119)(22 266 141 120)(23 267 142 81)(24 268 143 82)(25 269 144 83)(26 270 145 84)(27 271 146 85)(28 272 147 86)(29 273 148 87)(30 274 149 88)(31 275 150 89)(32 276 151 90)(33 277 152 91)(34 278 153 92)(35 279 154 93)(36 280 155 94)(37 241 156 95)(38 242 157 96)(39 243 158 97)(40 244 159 98)(41 225 169 318)(42 226 170 319)(43 227 171 320)(44 228 172 281)(45 229 173 282)(46 230 174 283)(47 231 175 284)(48 232 176 285)(49 233 177 286)(50 234 178 287)(51 235 179 288)(52 236 180 289)(53 237 181 290)(54 238 182 291)(55 239 183 292)(56 240 184 293)(57 201 185 294)(58 202 186 295)(59 203 187 296)(60 204 188 297)(61 205 189 298)(62 206 190 299)(63 207 191 300)(64 208 192 301)(65 209 193 302)(66 210 194 303)(67 211 195 304)(68 212 196 305)(69 213 197 306)(70 214 198 307)(71 215 199 308)(72 216 200 309)(73 217 161 310)(74 218 162 311)(75 219 163 312)(76 220 164 313)(77 221 165 314)(78 222 166 315)(79 223 167 316)(80 224 168 317)
(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)| (1,308)(2,309)(3,310)(4,311)(5,312)(6,313)(7,314)(8,315)(9,316)(10,317)(11,318)(12,319)(13,320)(14,281)(15,282)(16,283)(17,284)(18,285)(19,286)(20,287)(21,288)(22,289)(23,290)(24,291)(25,292)(26,293)(27,294)(28,295)(29,296)(30,297)(31,298)(32,299)(33,300)(34,301)(35,302)(36,303)(37,304)(38,305)(39,306)(40,307)(41,255)(42,256)(43,257)(44,258)(45,259)(46,260)(47,261)(48,262)(49,263)(50,264)(51,265)(52,266)(53,267)(54,268)(55,269)(56,270)(57,271)(58,272)(59,273)(60,274)(61,275)(62,276)(63,277)(64,278)(65,279)(66,280)(67,241)(68,242)(69,243)(70,244)(71,245)(72,246)(73,247)(74,248)(75,249)(76,250)(77,251)(78,252)(79,253)(80,254)(81,181)(82,182)(83,183)(84,184)(85,185)(86,186)(87,187)(88,188)(89,189)(90,190)(91,191)(92,192)(93,193)(94,194)(95,195)(96,196)(97,197)(98,198)(99,199)(100,200)(101,161)(102,162)(103,163)(104,164)(105,165)(106,166)(107,167)(108,168)(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)(117,177)(118,178)(119,179)(120,180)(121,216)(122,217)(123,218)(124,219)(125,220)(126,221)(127,222)(128,223)(129,224)(130,225)(131,226)(132,227)(133,228)(134,229)(135,230)(136,231)(137,232)(138,233)(139,234)(140,235)(141,236)(142,237)(143,238)(144,239)(145,240)(146,201)(147,202)(148,203)(149,204)(150,205)(151,206)(152,207)(153,208)(154,209)(155,210)(156,211)(157,212)(158,213)(159,214)(160,215), (1,245,160,99)(2,246,121,100)(3,247,122,101)(4,248,123,102)(5,249,124,103)(6,250,125,104)(7,251,126,105)(8,252,127,106)(9,253,128,107)(10,254,129,108)(11,255,130,109)(12,256,131,110)(13,257,132,111)(14,258,133,112)(15,259,134,113)(16,260,135,114)(17,261,136,115)(18,262,137,116)(19,263,138,117)(20,264,139,118)(21,265,140,119)(22,266,141,120)(23,267,142,81)(24,268,143,82)(25,269,144,83)(26,270,145,84)(27,271,146,85)(28,272,147,86)(29,273,148,87)(30,274,149,88)(31,275,150,89)(32,276,151,90)(33,277,152,91)(34,278,153,92)(35,279,154,93)(36,280,155,94)(37,241,156,95)(38,242,157,96)(39,243,158,97)(40,244,159,98)(41,225,169,318)(42,226,170,319)(43,227,171,320)(44,228,172,281)(45,229,173,282)(46,230,174,283)(47,231,175,284)(48,232,176,285)(49,233,177,286)(50,234,178,287)(51,235,179,288)(52,236,180,289)(53,237,181,290)(54,238,182,291)(55,239,183,292)(56,240,184,293)(57,201,185,294)(58,202,186,295)(59,203,187,296)(60,204,188,297)(61,205,189,298)(62,206,190,299)(63,207,191,300)(64,208,192,301)(65,209,193,302)(66,210,194,303)(67,211,195,304)(68,212,196,305)(69,213,197,306)(70,214,198,307)(71,215,199,308)(72,216,200,309)(73,217,161,310)(74,218,162,311)(75,219,163,312)(76,220,164,313)(77,221,165,314)(78,222,166,315)(79,223,167,316)(80,224,168,317), (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( (1,308)(2,309)(3,310)(4,311)(5,312)(6,313)(7,314)(8,315)(9,316)(10,317)(11,318)(12,319)(13,320)(14,281)(15,282)(16,283)(17,284)(18,285)(19,286)(20,287)(21,288)(22,289)(23,290)(24,291)(25,292)(26,293)(27,294)(28,295)(29,296)(30,297)(31,298)(32,299)(33,300)(34,301)(35,302)(36,303)(37,304)(38,305)(39,306)(40,307)(41,255)(42,256)(43,257)(44,258)(45,259)(46,260)(47,261)(48,262)(49,263)(50,264)(51,265)(52,266)(53,267)(54,268)(55,269)(56,270)(57,271)(58,272)(59,273)(60,274)(61,275)(62,276)(63,277)(64,278)(65,279)(66,280)(67,241)(68,242)(69,243)(70,244)(71,245)(72,246)(73,247)(74,248)(75,249)(76,250)(77,251)(78,252)(79,253)(80,254)(81,181)(82,182)(83,183)(84,184)(85,185)(86,186)(87,187)(88,188)(89,189)(90,190)(91,191)(92,192)(93,193)(94,194)(95,195)(96,196)(97,197)(98,198)(99,199)(100,200)(101,161)(102,162)(103,163)(104,164)(105,165)(106,166)(107,167)(108,168)(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)(117,177)(118,178)(119,179)(120,180)(121,216)(122,217)(123,218)(124,219)(125,220)(126,221)(127,222)(128,223)(129,224)(130,225)(131,226)(132,227)(133,228)(134,229)(135,230)(136,231)(137,232)(138,233)(139,234)(140,235)(141,236)(142,237)(143,238)(144,239)(145,240)(146,201)(147,202)(148,203)(149,204)(150,205)(151,206)(152,207)(153,208)(154,209)(155,210)(156,211)(157,212)(158,213)(159,214)(160,215), (1,245,160,99)(2,246,121,100)(3,247,122,101)(4,248,123,102)(5,249,124,103)(6,250,125,104)(7,251,126,105)(8,252,127,106)(9,253,128,107)(10,254,129,108)(11,255,130,109)(12,256,131,110)(13,257,132,111)(14,258,133,112)(15,259,134,113)(16,260,135,114)(17,261,136,115)(18,262,137,116)(19,263,138,117)(20,264,139,118)(21,265,140,119)(22,266,141,120)(23,267,142,81)(24,268,143,82)(25,269,144,83)(26,270,145,84)(27,271,146,85)(28,272,147,86)(29,273,148,87)(30,274,149,88)(31,275,150,89)(32,276,151,90)(33,277,152,91)(34,278,153,92)(35,279,154,93)(36,280,155,94)(37,241,156,95)(38,242,157,96)(39,243,158,97)(40,244,159,98)(41,225,169,318)(42,226,170,319)(43,227,171,320)(44,228,172,281)(45,229,173,282)(46,230,174,283)(47,231,175,284)(48,232,176,285)(49,233,177,286)(50,234,178,287)(51,235,179,288)(52,236,180,289)(53,237,181,290)(54,238,182,291)(55,239,183,292)(56,240,184,293)(57,201,185,294)(58,202,186,295)(59,203,187,296)(60,204,188,297)(61,205,189,298)(62,206,190,299)(63,207,191,300)(64,208,192,301)(65,209,193,302)(66,210,194,303)(67,211,195,304)(68,212,196,305)(69,213,197,306)(70,214,198,307)(71,215,199,308)(72,216,200,309)(73,217,161,310)(74,218,162,311)(75,219,163,312)(76,220,164,313)(77,221,165,314)(78,222,166,315)(79,223,167,316)(80,224,168,317), (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([(1,308),(2,309),(3,310),(4,311),(5,312),(6,313),(7,314),(8,315),(9,316),(10,317),(11,318),(12,319),(13,320),(14,281),(15,282),(16,283),(17,284),(18,285),(19,286),(20,287),(21,288),(22,289),(23,290),(24,291),(25,292),(26,293),(27,294),(28,295),(29,296),(30,297),(31,298),(32,299),(33,300),(34,301),(35,302),(36,303),(37,304),(38,305),(39,306),(40,307),(41,255),(42,256),(43,257),(44,258),(45,259),(46,260),(47,261),(48,262),(49,263),(50,264),(51,265),(52,266),(53,267),(54,268),(55,269),(56,270),(57,271),(58,272),(59,273),(60,274),(61,275),(62,276),(63,277),(64,278),(65,279),(66,280),(67,241),(68,242),(69,243),(70,244),(71,245),(72,246),(73,247),(74,248),(75,249),(76,250),(77,251),(78,252),(79,253),(80,254),(81,181),(82,182),(83,183),(84,184),(85,185),(86,186),(87,187),(88,188),(89,189),(90,190),(91,191),(92,192),(93,193),(94,194),(95,195),(96,196),(97,197),(98,198),(99,199),(100,200),(101,161),(102,162),(103,163),(104,164),(105,165),(106,166),(107,167),(108,168),(109,169),(110,170),(111,171),(112,172),(113,173),(114,174),(115,175),(116,176),(117,177),(118,178),(119,179),(120,180),(121,216),(122,217),(123,218),(124,219),(125,220),(126,221),(127,222),(128,223),(129,224),(130,225),(131,226),(132,227),(133,228),(134,229),(135,230),(136,231),(137,232),(138,233),(139,234),(140,235),(141,236),(142,237),(143,238),(144,239),(145,240),(146,201),(147,202),(148,203),(149,204),(150,205),(151,206),(152,207),(153,208),(154,209),(155,210),(156,211),(157,212),(158,213),(159,214),(160,215)], [(1,245,160,99),(2,246,121,100),(3,247,122,101),(4,248,123,102),(5,249,124,103),(6,250,125,104),(7,251,126,105),(8,252,127,106),(9,253,128,107),(10,254,129,108),(11,255,130,109),(12,256,131,110),(13,257,132,111),(14,258,133,112),(15,259,134,113),(16,260,135,114),(17,261,136,115),(18,262,137,116),(19,263,138,117),(20,264,139,118),(21,265,140,119),(22,266,141,120),(23,267,142,81),(24,268,143,82),(25,269,144,83),(26,270,145,84),(27,271,146,85),(28,272,147,86),(29,273,148,87),(30,274,149,88),(31,275,150,89),(32,276,151,90),(33,277,152,91),(34,278,153,92),(35,279,154,93),(36,280,155,94),(37,241,156,95),(38,242,157,96),(39,243,158,97),(40,244,159,98),(41,225,169,318),(42,226,170,319),(43,227,171,320),(44,228,172,281),(45,229,173,282),(46,230,174,283),(47,231,175,284),(48,232,176,285),(49,233,177,286),(50,234,178,287),(51,235,179,288),(52,236,180,289),(53,237,181,290),(54,238,182,291),(55,239,183,292),(56,240,184,293),(57,201,185,294),(58,202,186,295),(59,203,187,296),(60,204,188,297),(61,205,189,298),(62,206,190,299),(63,207,191,300),(64,208,192,301),(65,209,193,302),(66,210,194,303),(67,211,195,304),(68,212,196,305),(69,213,197,306),(70,214,198,307),(71,215,199,308),(72,216,200,309),(73,217,161,310),(74,218,162,311),(75,219,163,312),(76,220,164,313),(77,221,165,314),(78,222,166,315),(79,223,167,316),(80,224,168,317)], [(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

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