direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C4⋊C16, C4⋊C80, C20⋊5C16, C40.23Q8, C40.108D4, C42.7C20, C10.15M5(2), C20.61M4(2), C8.7(C5×Q8), (C4×C40).4C2, (C2×C80).4C2, C2.2(C2×C80), (C2×C4).4C40, (C2×C8).7C20, (C4×C8).2C10, C8.28(C5×D4), (C2×C16).2C10, (C4×C20).39C4, (C2×C40).41C4, (C2×C20).22C8, C20.98(C4⋊C4), C10.23(C4⋊C8), C10.22(C2×C16), C2.3(C5×M5(2)), C22.10(C2×C40), C4.11(C5×M4(2)), (C2×C40).452C22, C2.2(C5×C4⋊C8), C4.18(C5×C4⋊C4), (C2×C10).69(C2×C8), (C2×C4).84(C2×C20), (C2×C8).106(C2×C10), (C2×C20).519(C2×C4), SmallGroup(320,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊C16
G = < a,b,c | a5=b4=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C4⋊C16
►Regular action on 320 points
Generators in S320
(1 285 193 305 158)(2 286 194 306 159)(3 287 195 307 160)(4 288 196 308 145)(5 273 197 309 146)(6 274 198 310 147)(7 275 199 311 148)(8 276 200 312 149)(9 277 201 313 150)(10 278 202 314 151)(11 279 203 315 152)(12 280 204 316 153)(13 281 205 317 154)(14 282 206 318 155)(15 283 207 319 156)(16 284 208 320 157)(17 302 117 57 240)(18 303 118 58 225)(19 304 119 59 226)(20 289 120 60 227)(21 290 121 61 228)(22 291 122 62 229)(23 292 123 63 230)(24 293 124 64 231)(25 294 125 49 232)(26 295 126 50 233)(27 296 127 51 234)(28 297 128 52 235)(29 298 113 53 236)(30 299 114 54 237)(31 300 115 55 238)(32 301 116 56 239)(33 81 270 179 144)(34 82 271 180 129)(35 83 272 181 130)(36 84 257 182 131)(37 85 258 183 132)(38 86 259 184 133)(39 87 260 185 134)(40 88 261 186 135)(41 89 262 187 136)(42 90 263 188 137)(43 91 264 189 138)(44 92 265 190 139)(45 93 266 191 140)(46 94 267 192 141)(47 95 268 177 142)(48 96 269 178 143)(65 243 219 175 106)(66 244 220 176 107)(67 245 221 161 108)(68 246 222 162 109)(69 247 223 163 110)(70 248 224 164 111)(71 249 209 165 112)(72 250 210 166 97)(73 251 211 167 98)(74 252 212 168 99)(75 253 213 169 100)(76 254 214 170 101)(77 255 215 171 102)(78 256 216 172 103)(79 241 217 173 104)(80 242 218 174 105)
(1 35 105 128)(2 113 106 36)(3 37 107 114)(4 115 108 38)(5 39 109 116)(6 117 110 40)(7 41 111 118)(8 119 112 42)(9 43 97 120)(10 121 98 44)(11 45 99 122)(12 123 100 46)(13 47 101 124)(14 125 102 48)(15 33 103 126)(16 127 104 34)(17 223 186 310)(18 311 187 224)(19 209 188 312)(20 313 189 210)(21 211 190 314)(22 315 191 212)(23 213 192 316)(24 317 177 214)(25 215 178 318)(26 319 179 216)(27 217 180 320)(28 305 181 218)(29 219 182 306)(30 307 183 220)(31 221 184 308)(32 309 185 222)(49 77 96 282)(50 283 81 78)(51 79 82 284)(52 285 83 80)(53 65 84 286)(54 287 85 66)(55 67 86 288)(56 273 87 68)(57 69 88 274)(58 275 89 70)(59 71 90 276)(60 277 91 72)(61 73 92 278)(62 279 93 74)(63 75 94 280)(64 281 95 76)(129 157 296 173)(130 174 297 158)(131 159 298 175)(132 176 299 160)(133 145 300 161)(134 162 301 146)(135 147 302 163)(136 164 303 148)(137 149 304 165)(138 166 289 150)(139 151 290 167)(140 168 291 152)(141 153 292 169)(142 170 293 154)(143 155 294 171)(144 172 295 156)(193 272 242 235)(194 236 243 257)(195 258 244 237)(196 238 245 259)(197 260 246 239)(198 240 247 261)(199 262 248 225)(200 226 249 263)(201 264 250 227)(202 228 251 265)(203 266 252 229)(204 230 253 267)(205 268 254 231)(206 232 255 269)(207 270 256 233)(208 234 241 271)
(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)| (1,285,193,305,158)(2,286,194,306,159)(3,287,195,307,160)(4,288,196,308,145)(5,273,197,309,146)(6,274,198,310,147)(7,275,199,311,148)(8,276,200,312,149)(9,277,201,313,150)(10,278,202,314,151)(11,279,203,315,152)(12,280,204,316,153)(13,281,205,317,154)(14,282,206,318,155)(15,283,207,319,156)(16,284,208,320,157)(17,302,117,57,240)(18,303,118,58,225)(19,304,119,59,226)(20,289,120,60,227)(21,290,121,61,228)(22,291,122,62,229)(23,292,123,63,230)(24,293,124,64,231)(25,294,125,49,232)(26,295,126,50,233)(27,296,127,51,234)(28,297,128,52,235)(29,298,113,53,236)(30,299,114,54,237)(31,300,115,55,238)(32,301,116,56,239)(33,81,270,179,144)(34,82,271,180,129)(35,83,272,181,130)(36,84,257,182,131)(37,85,258,183,132)(38,86,259,184,133)(39,87,260,185,134)(40,88,261,186,135)(41,89,262,187,136)(42,90,263,188,137)(43,91,264,189,138)(44,92,265,190,139)(45,93,266,191,140)(46,94,267,192,141)(47,95,268,177,142)(48,96,269,178,143)(65,243,219,175,106)(66,244,220,176,107)(67,245,221,161,108)(68,246,222,162,109)(69,247,223,163,110)(70,248,224,164,111)(71,249,209,165,112)(72,250,210,166,97)(73,251,211,167,98)(74,252,212,168,99)(75,253,213,169,100)(76,254,214,170,101)(77,255,215,171,102)(78,256,216,172,103)(79,241,217,173,104)(80,242,218,174,105), (1,35,105,128)(2,113,106,36)(3,37,107,114)(4,115,108,38)(5,39,109,116)(6,117,110,40)(7,41,111,118)(8,119,112,42)(9,43,97,120)(10,121,98,44)(11,45,99,122)(12,123,100,46)(13,47,101,124)(14,125,102,48)(15,33,103,126)(16,127,104,34)(17,223,186,310)(18,311,187,224)(19,209,188,312)(20,313,189,210)(21,211,190,314)(22,315,191,212)(23,213,192,316)(24,317,177,214)(25,215,178,318)(26,319,179,216)(27,217,180,320)(28,305,181,218)(29,219,182,306)(30,307,183,220)(31,221,184,308)(32,309,185,222)(49,77,96,282)(50,283,81,78)(51,79,82,284)(52,285,83,80)(53,65,84,286)(54,287,85,66)(55,67,86,288)(56,273,87,68)(57,69,88,274)(58,275,89,70)(59,71,90,276)(60,277,91,72)(61,73,92,278)(62,279,93,74)(63,75,94,280)(64,281,95,76)(129,157,296,173)(130,174,297,158)(131,159,298,175)(132,176,299,160)(133,145,300,161)(134,162,301,146)(135,147,302,163)(136,164,303,148)(137,149,304,165)(138,166,289,150)(139,151,290,167)(140,168,291,152)(141,153,292,169)(142,170,293,154)(143,155,294,171)(144,172,295,156)(193,272,242,235)(194,236,243,257)(195,258,244,237)(196,238,245,259)(197,260,246,239)(198,240,247,261)(199,262,248,225)(200,226,249,263)(201,264,250,227)(202,228,251,265)(203,266,252,229)(204,230,253,267)(205,268,254,231)(206,232,255,269)(207,270,256,233)(208,234,241,271), (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( (1,285,193,305,158)(2,286,194,306,159)(3,287,195,307,160)(4,288,196,308,145)(5,273,197,309,146)(6,274,198,310,147)(7,275,199,311,148)(8,276,200,312,149)(9,277,201,313,150)(10,278,202,314,151)(11,279,203,315,152)(12,280,204,316,153)(13,281,205,317,154)(14,282,206,318,155)(15,283,207,319,156)(16,284,208,320,157)(17,302,117,57,240)(18,303,118,58,225)(19,304,119,59,226)(20,289,120,60,227)(21,290,121,61,228)(22,291,122,62,229)(23,292,123,63,230)(24,293,124,64,231)(25,294,125,49,232)(26,295,126,50,233)(27,296,127,51,234)(28,297,128,52,235)(29,298,113,53,236)(30,299,114,54,237)(31,300,115,55,238)(32,301,116,56,239)(33,81,270,179,144)(34,82,271,180,129)(35,83,272,181,130)(36,84,257,182,131)(37,85,258,183,132)(38,86,259,184,133)(39,87,260,185,134)(40,88,261,186,135)(41,89,262,187,136)(42,90,263,188,137)(43,91,264,189,138)(44,92,265,190,139)(45,93,266,191,140)(46,94,267,192,141)(47,95,268,177,142)(48,96,269,178,143)(65,243,219,175,106)(66,244,220,176,107)(67,245,221,161,108)(68,246,222,162,109)(69,247,223,163,110)(70,248,224,164,111)(71,249,209,165,112)(72,250,210,166,97)(73,251,211,167,98)(74,252,212,168,99)(75,253,213,169,100)(76,254,214,170,101)(77,255,215,171,102)(78,256,216,172,103)(79,241,217,173,104)(80,242,218,174,105), (1,35,105,128)(2,113,106,36)(3,37,107,114)(4,115,108,38)(5,39,109,116)(6,117,110,40)(7,41,111,118)(8,119,112,42)(9,43,97,120)(10,121,98,44)(11,45,99,122)(12,123,100,46)(13,47,101,124)(14,125,102,48)(15,33,103,126)(16,127,104,34)(17,223,186,310)(18,311,187,224)(19,209,188,312)(20,313,189,210)(21,211,190,314)(22,315,191,212)(23,213,192,316)(24,317,177,214)(25,215,178,318)(26,319,179,216)(27,217,180,320)(28,305,181,218)(29,219,182,306)(30,307,183,220)(31,221,184,308)(32,309,185,222)(49,77,96,282)(50,283,81,78)(51,79,82,284)(52,285,83,80)(53,65,84,286)(54,287,85,66)(55,67,86,288)(56,273,87,68)(57,69,88,274)(58,275,89,70)(59,71,90,276)(60,277,91,72)(61,73,92,278)(62,279,93,74)(63,75,94,280)(64,281,95,76)(129,157,296,173)(130,174,297,158)(131,159,298,175)(132,176,299,160)(133,145,300,161)(134,162,301,146)(135,147,302,163)(136,164,303,148)(137,149,304,165)(138,166,289,150)(139,151,290,167)(140,168,291,152)(141,153,292,169)(142,170,293,154)(143,155,294,171)(144,172,295,156)(193,272,242,235)(194,236,243,257)(195,258,244,237)(196,238,245,259)(197,260,246,239)(198,240,247,261)(199,262,248,225)(200,226,249,263)(201,264,250,227)(202,228,251,265)(203,266,252,229)(204,230,253,267)(205,268,254,231)(206,232,255,269)(207,270,256,233)(208,234,241,271), (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([[(1,285,193,305,158),(2,286,194,306,159),(3,287,195,307,160),(4,288,196,308,145),(5,273,197,309,146),(6,274,198,310,147),(7,275,199,311,148),(8,276,200,312,149),(9,277,201,313,150),(10,278,202,314,151),(11,279,203,315,152),(12,280,204,316,153),(13,281,205,317,154),(14,282,206,318,155),(15,283,207,319,156),(16,284,208,320,157),(17,302,117,57,240),(18,303,118,58,225),(19,304,119,59,226),(20,289,120,60,227),(21,290,121,61,228),(22,291,122,62,229),(23,292,123,63,230),(24,293,124,64,231),(25,294,125,49,232),(26,295,126,50,233),(27,296,127,51,234),(28,297,128,52,235),(29,298,113,53,236),(30,299,114,54,237),(31,300,115,55,238),(32,301,116,56,239),(33,81,270,179,144),(34,82,271,180,129),(35,83,272,181,130),(36,84,257,182,131),(37,85,258,183,132),(38,86,259,184,133),(39,87,260,185,134),(40,88,261,186,135),(41,89,262,187,136),(42,90,263,188,137),(43,91,264,189,138),(44,92,265,190,139),(45,93,266,191,140),(46,94,267,192,141),(47,95,268,177,142),(48,96,269,178,143),(65,243,219,175,106),(66,244,220,176,107),(67,245,221,161,108),(68,246,222,162,109),(69,247,223,163,110),(70,248,224,164,111),(71,249,209,165,112),(72,250,210,166,97),(73,251,211,167,98),(74,252,212,168,99),(75,253,213,169,100),(76,254,214,170,101),(77,255,215,171,102),(78,256,216,172,103),(79,241,217,173,104),(80,242,218,174,105)], [(1,35,105,128),(2,113,106,36),(3,37,107,114),(4,115,108,38),(5,39,109,116),(6,117,110,40),(7,41,111,118),(8,119,112,42),(9,43,97,120),(10,121,98,44),(11,45,99,122),(12,123,100,46),(13,47,101,124),(14,125,102,48),(15,33,103,126),(16,127,104,34),(17,223,186,310),(18,311,187,224),(19,209,188,312),(20,313,189,210),(21,211,190,314),(22,315,191,212),(23,213,192,316),(24,317,177,214),(25,215,178,318),(26,319,179,216),(27,217,180,320),(28,305,181,218),(29,219,182,306),(30,307,183,220),(31,221,184,308),(32,309,185,222),(49,77,96,282),(50,283,81,78),(51,79,82,284),(52,285,83,80),(53,65,84,286),(54,287,85,66),(55,67,86,288),(56,273,87,68),(57,69,88,274),(58,275,89,70),(59,71,90,276),(60,277,91,72),(61,73,92,278),(62,279,93,74),(63,75,94,280),(64,281,95,76),(129,157,296,173),(130,174,297,158),(131,159,298,175),(132,176,299,160),(133,145,300,161),(134,162,301,146),(135,147,302,163),(136,164,303,148),(137,149,304,165),(138,166,289,150),(139,151,290,167),(140,168,291,152),(141,153,292,169),(142,170,293,154),(143,155,294,171),(144,172,295,156),(193,272,242,235),(194,236,243,257),(195,258,244,237),(196,238,245,259),(197,260,246,239),(198,240,247,261),(199,262,248,225),(200,226,249,263),(201,264,250,227),(202,228,251,265),(203,266,252,229),(204,230,253,267),(205,268,254,231),(206,232,255,269),(207,270,256,233),(208,234,241,271)], [(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)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10L | 16A | ··· | 16P | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF | 40AG | ··· | 40AV | 80A | ··· | 80BL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C16 | C20 | C20 | C40 | C80 | D4 | Q8 | M4(2) | M5(2) | C5×D4 | C5×Q8 | C5×M4(2) | C5×M5(2) |
| kernel | C5×C4⋊C16 | C4×C40 | C2×C80 | C4×C20 | C2×C40 | C4⋊C16 | C2×C20 | C4×C8 | C2×C16 | C20 | C42 | C2×C8 | C2×C4 | C4 | C40 | C40 | C20 | C10 | C8 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 4 | 8 | 16 | 8 | 8 | 32 | 64 | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 |
Matrix representation of C5×C4⋊C16 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 240 | 0 | 0 |
| 0 | 222 | 226 |
| 0 | 233 | 19 |
| 197 | 0 | 0 |
| 0 | 134 | 214 |
| 0 | 173 | 107 |
G:=sub<GL(3,GF(241))| [1,0,0,0,98,0,0,0,98],[240,0,0,0,222,233,0,226,19],[197,0,0,0,134,173,0,214,107] >;
C5×C4⋊C16 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes C_{16} % in TeX
G:=Group("C5xC4:C16"); // GroupNames label
G:=SmallGroup(320,168);
// by ID
G=gap.SmallGroup(320,168);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,148,102,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^4=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export