direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C16⋊5C4, C80⋊19C4, C16⋊5C20, C8.2C40, C40.14C8, C42.4C20, C20.64C42, C10.13M5(2), C4.6(C4×C20), C2.3(C4×C40), (C2×C4).2C40, C10.23(C4×C8), (C2×C40).55C4, (C4×C8).14C10, (C4×C40).32C2, (C2×C80).17C2, C4.12(C2×C40), (C4×C20).24C4, (C2×C16).7C10, C8.21(C2×C20), C20.86(C2×C8), (C2×C20).15C8, (C2×C8).12C20, C40.131(C2×C4), C22.8(C2×C40), C2.1(C5×M5(2)), (C2×C40).450C22, (C2×C10).67(C2×C8), (C2×C4).82(C2×C20), (C2×C8).104(C2×C10), (C2×C20).517(C2×C4), SmallGroup(320,151)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C16⋊5C4
G = < a,b,c | a5=b16=c4=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C5×C16⋊5C4
►Regular action on 320 points
Generators in S320
(1 29 120 182 150)(2 30 121 183 151)(3 31 122 184 152)(4 32 123 185 153)(5 17 124 186 154)(6 18 125 187 155)(7 19 126 188 156)(8 20 127 189 157)(9 21 128 190 158)(10 22 113 191 159)(11 23 114 192 160)(12 24 115 177 145)(13 25 116 178 146)(14 26 117 179 147)(15 27 118 180 148)(16 28 119 181 149)(33 109 75 88 319)(34 110 76 89 320)(35 111 77 90 305)(36 112 78 91 306)(37 97 79 92 307)(38 98 80 93 308)(39 99 65 94 309)(40 100 66 95 310)(41 101 67 96 311)(42 102 68 81 312)(43 103 69 82 313)(44 104 70 83 314)(45 105 71 84 315)(46 106 72 85 316)(47 107 73 86 317)(48 108 74 87 318)(49 290 143 195 163)(50 291 144 196 164)(51 292 129 197 165)(52 293 130 198 166)(53 294 131 199 167)(54 295 132 200 168)(55 296 133 201 169)(56 297 134 202 170)(57 298 135 203 171)(58 299 136 204 172)(59 300 137 205 173)(60 301 138 206 174)(61 302 139 207 175)(62 303 140 208 176)(63 304 141 193 161)(64 289 142 194 162)(209 256 262 276 225)(210 241 263 277 226)(211 242 264 278 227)(212 243 265 279 228)(213 244 266 280 229)(214 245 267 281 230)(215 246 268 282 231)(216 247 269 283 232)(217 248 270 284 233)(218 249 271 285 234)(219 250 272 286 235)(220 251 257 287 236)(221 252 258 288 237)(222 253 259 273 238)(223 254 260 274 239)(224 255 261 275 240)
(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)
(1 54 219 78)(2 63 220 71)(3 56 221 80)(4 49 222 73)(5 58 223 66)(6 51 224 75)(7 60 209 68)(8 53 210 77)(9 62 211 70)(10 55 212 79)(11 64 213 72)(12 57 214 65)(13 50 215 74)(14 59 216 67)(15 52 217 76)(16 61 218 69)(17 299 254 95)(18 292 255 88)(19 301 256 81)(20 294 241 90)(21 303 242 83)(22 296 243 92)(23 289 244 85)(24 298 245 94)(25 291 246 87)(26 300 247 96)(27 293 248 89)(28 302 249 82)(29 295 250 91)(30 304 251 84)(31 297 252 93)(32 290 253 86)(33 187 197 275)(34 180 198 284)(35 189 199 277)(36 182 200 286)(37 191 201 279)(38 184 202 288)(39 177 203 281)(40 186 204 274)(41 179 205 283)(42 188 206 276)(43 181 207 285)(44 190 208 278)(45 183 193 287)(46 192 194 280)(47 185 195 273)(48 178 196 282)(97 159 169 228)(98 152 170 237)(99 145 171 230)(100 154 172 239)(101 147 173 232)(102 156 174 225)(103 149 175 234)(104 158 176 227)(105 151 161 236)(106 160 162 229)(107 153 163 238)(108 146 164 231)(109 155 165 240)(110 148 166 233)(111 157 167 226)(112 150 168 235)(113 133 265 307)(114 142 266 316)(115 135 267 309)(116 144 268 318)(117 137 269 311)(118 130 270 320)(119 139 271 313)(120 132 272 306)(121 141 257 315)(122 134 258 308)(123 143 259 317)(124 136 260 310)(125 129 261 319)(126 138 262 312)(127 131 263 305)(128 140 264 314)
G:=sub<Sym(320)| (1,29,120,182,150)(2,30,121,183,151)(3,31,122,184,152)(4,32,123,185,153)(5,17,124,186,154)(6,18,125,187,155)(7,19,126,188,156)(8,20,127,189,157)(9,21,128,190,158)(10,22,113,191,159)(11,23,114,192,160)(12,24,115,177,145)(13,25,116,178,146)(14,26,117,179,147)(15,27,118,180,148)(16,28,119,181,149)(33,109,75,88,319)(34,110,76,89,320)(35,111,77,90,305)(36,112,78,91,306)(37,97,79,92,307)(38,98,80,93,308)(39,99,65,94,309)(40,100,66,95,310)(41,101,67,96,311)(42,102,68,81,312)(43,103,69,82,313)(44,104,70,83,314)(45,105,71,84,315)(46,106,72,85,316)(47,107,73,86,317)(48,108,74,87,318)(49,290,143,195,163)(50,291,144,196,164)(51,292,129,197,165)(52,293,130,198,166)(53,294,131,199,167)(54,295,132,200,168)(55,296,133,201,169)(56,297,134,202,170)(57,298,135,203,171)(58,299,136,204,172)(59,300,137,205,173)(60,301,138,206,174)(61,302,139,207,175)(62,303,140,208,176)(63,304,141,193,161)(64,289,142,194,162)(209,256,262,276,225)(210,241,263,277,226)(211,242,264,278,227)(212,243,265,279,228)(213,244,266,280,229)(214,245,267,281,230)(215,246,268,282,231)(216,247,269,283,232)(217,248,270,284,233)(218,249,271,285,234)(219,250,272,286,235)(220,251,257,287,236)(221,252,258,288,237)(222,253,259,273,238)(223,254,260,274,239)(224,255,261,275,240), (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), (1,54,219,78)(2,63,220,71)(3,56,221,80)(4,49,222,73)(5,58,223,66)(6,51,224,75)(7,60,209,68)(8,53,210,77)(9,62,211,70)(10,55,212,79)(11,64,213,72)(12,57,214,65)(13,50,215,74)(14,59,216,67)(15,52,217,76)(16,61,218,69)(17,299,254,95)(18,292,255,88)(19,301,256,81)(20,294,241,90)(21,303,242,83)(22,296,243,92)(23,289,244,85)(24,298,245,94)(25,291,246,87)(26,300,247,96)(27,293,248,89)(28,302,249,82)(29,295,250,91)(30,304,251,84)(31,297,252,93)(32,290,253,86)(33,187,197,275)(34,180,198,284)(35,189,199,277)(36,182,200,286)(37,191,201,279)(38,184,202,288)(39,177,203,281)(40,186,204,274)(41,179,205,283)(42,188,206,276)(43,181,207,285)(44,190,208,278)(45,183,193,287)(46,192,194,280)(47,185,195,273)(48,178,196,282)(97,159,169,228)(98,152,170,237)(99,145,171,230)(100,154,172,239)(101,147,173,232)(102,156,174,225)(103,149,175,234)(104,158,176,227)(105,151,161,236)(106,160,162,229)(107,153,163,238)(108,146,164,231)(109,155,165,240)(110,148,166,233)(111,157,167,226)(112,150,168,235)(113,133,265,307)(114,142,266,316)(115,135,267,309)(116,144,268,318)(117,137,269,311)(118,130,270,320)(119,139,271,313)(120,132,272,306)(121,141,257,315)(122,134,258,308)(123,143,259,317)(124,136,260,310)(125,129,261,319)(126,138,262,312)(127,131,263,305)(128,140,264,314)>;
G:=Group( (1,29,120,182,150)(2,30,121,183,151)(3,31,122,184,152)(4,32,123,185,153)(5,17,124,186,154)(6,18,125,187,155)(7,19,126,188,156)(8,20,127,189,157)(9,21,128,190,158)(10,22,113,191,159)(11,23,114,192,160)(12,24,115,177,145)(13,25,116,178,146)(14,26,117,179,147)(15,27,118,180,148)(16,28,119,181,149)(33,109,75,88,319)(34,110,76,89,320)(35,111,77,90,305)(36,112,78,91,306)(37,97,79,92,307)(38,98,80,93,308)(39,99,65,94,309)(40,100,66,95,310)(41,101,67,96,311)(42,102,68,81,312)(43,103,69,82,313)(44,104,70,83,314)(45,105,71,84,315)(46,106,72,85,316)(47,107,73,86,317)(48,108,74,87,318)(49,290,143,195,163)(50,291,144,196,164)(51,292,129,197,165)(52,293,130,198,166)(53,294,131,199,167)(54,295,132,200,168)(55,296,133,201,169)(56,297,134,202,170)(57,298,135,203,171)(58,299,136,204,172)(59,300,137,205,173)(60,301,138,206,174)(61,302,139,207,175)(62,303,140,208,176)(63,304,141,193,161)(64,289,142,194,162)(209,256,262,276,225)(210,241,263,277,226)(211,242,264,278,227)(212,243,265,279,228)(213,244,266,280,229)(214,245,267,281,230)(215,246,268,282,231)(216,247,269,283,232)(217,248,270,284,233)(218,249,271,285,234)(219,250,272,286,235)(220,251,257,287,236)(221,252,258,288,237)(222,253,259,273,238)(223,254,260,274,239)(224,255,261,275,240), (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), (1,54,219,78)(2,63,220,71)(3,56,221,80)(4,49,222,73)(5,58,223,66)(6,51,224,75)(7,60,209,68)(8,53,210,77)(9,62,211,70)(10,55,212,79)(11,64,213,72)(12,57,214,65)(13,50,215,74)(14,59,216,67)(15,52,217,76)(16,61,218,69)(17,299,254,95)(18,292,255,88)(19,301,256,81)(20,294,241,90)(21,303,242,83)(22,296,243,92)(23,289,244,85)(24,298,245,94)(25,291,246,87)(26,300,247,96)(27,293,248,89)(28,302,249,82)(29,295,250,91)(30,304,251,84)(31,297,252,93)(32,290,253,86)(33,187,197,275)(34,180,198,284)(35,189,199,277)(36,182,200,286)(37,191,201,279)(38,184,202,288)(39,177,203,281)(40,186,204,274)(41,179,205,283)(42,188,206,276)(43,181,207,285)(44,190,208,278)(45,183,193,287)(46,192,194,280)(47,185,195,273)(48,178,196,282)(97,159,169,228)(98,152,170,237)(99,145,171,230)(100,154,172,239)(101,147,173,232)(102,156,174,225)(103,149,175,234)(104,158,176,227)(105,151,161,236)(106,160,162,229)(107,153,163,238)(108,146,164,231)(109,155,165,240)(110,148,166,233)(111,157,167,226)(112,150,168,235)(113,133,265,307)(114,142,266,316)(115,135,267,309)(116,144,268,318)(117,137,269,311)(118,130,270,320)(119,139,271,313)(120,132,272,306)(121,141,257,315)(122,134,258,308)(123,143,259,317)(124,136,260,310)(125,129,261,319)(126,138,262,312)(127,131,263,305)(128,140,264,314) );
G=PermutationGroup([[(1,29,120,182,150),(2,30,121,183,151),(3,31,122,184,152),(4,32,123,185,153),(5,17,124,186,154),(6,18,125,187,155),(7,19,126,188,156),(8,20,127,189,157),(9,21,128,190,158),(10,22,113,191,159),(11,23,114,192,160),(12,24,115,177,145),(13,25,116,178,146),(14,26,117,179,147),(15,27,118,180,148),(16,28,119,181,149),(33,109,75,88,319),(34,110,76,89,320),(35,111,77,90,305),(36,112,78,91,306),(37,97,79,92,307),(38,98,80,93,308),(39,99,65,94,309),(40,100,66,95,310),(41,101,67,96,311),(42,102,68,81,312),(43,103,69,82,313),(44,104,70,83,314),(45,105,71,84,315),(46,106,72,85,316),(47,107,73,86,317),(48,108,74,87,318),(49,290,143,195,163),(50,291,144,196,164),(51,292,129,197,165),(52,293,130,198,166),(53,294,131,199,167),(54,295,132,200,168),(55,296,133,201,169),(56,297,134,202,170),(57,298,135,203,171),(58,299,136,204,172),(59,300,137,205,173),(60,301,138,206,174),(61,302,139,207,175),(62,303,140,208,176),(63,304,141,193,161),(64,289,142,194,162),(209,256,262,276,225),(210,241,263,277,226),(211,242,264,278,227),(212,243,265,279,228),(213,244,266,280,229),(214,245,267,281,230),(215,246,268,282,231),(216,247,269,283,232),(217,248,270,284,233),(218,249,271,285,234),(219,250,272,286,235),(220,251,257,287,236),(221,252,258,288,237),(222,253,259,273,238),(223,254,260,274,239),(224,255,261,275,240)], [(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)], [(1,54,219,78),(2,63,220,71),(3,56,221,80),(4,49,222,73),(5,58,223,66),(6,51,224,75),(7,60,209,68),(8,53,210,77),(9,62,211,70),(10,55,212,79),(11,64,213,72),(12,57,214,65),(13,50,215,74),(14,59,216,67),(15,52,217,76),(16,61,218,69),(17,299,254,95),(18,292,255,88),(19,301,256,81),(20,294,241,90),(21,303,242,83),(22,296,243,92),(23,289,244,85),(24,298,245,94),(25,291,246,87),(26,300,247,96),(27,293,248,89),(28,302,249,82),(29,295,250,91),(30,304,251,84),(31,297,252,93),(32,290,253,86),(33,187,197,275),(34,180,198,284),(35,189,199,277),(36,182,200,286),(37,191,201,279),(38,184,202,288),(39,177,203,281),(40,186,204,274),(41,179,205,283),(42,188,206,276),(43,181,207,285),(44,190,208,278),(45,183,193,287),(46,192,194,280),(47,185,195,273),(48,178,196,282),(97,159,169,228),(98,152,170,237),(99,145,171,230),(100,154,172,239),(101,147,173,232),(102,156,174,225),(103,149,175,234),(104,158,176,227),(105,151,161,236),(106,160,162,229),(107,153,163,238),(108,146,164,231),(109,155,165,240),(110,148,166,233),(111,157,167,226),(112,150,168,235),(113,133,265,307),(114,142,266,316),(115,135,267,309),(116,144,268,318),(117,137,269,311),(118,130,270,320),(119,139,271,313),(120,132,272,306),(121,141,257,315),(122,134,258,308),(123,143,259,317),(124,136,260,310),(125,129,261,319),(126,138,262,312),(127,131,263,305),(128,140,264,314)]])
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 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C5 | C8 | C8 | C10 | C10 | C20 | C20 | C20 | C40 | C40 | M5(2) | C5×M5(2) |
| kernel | C5×C16⋊5C4 | C4×C40 | C2×C80 | C80 | C4×C20 | C2×C40 | C16⋊5C4 | C40 | C2×C20 | C4×C8 | C2×C16 | C16 | C42 | C2×C8 | C8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 32 | 8 | 8 | 32 | 32 | 8 | 32 |
Matrix representation of C5×C16⋊5C4 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 205 | 0 |
| 0 | 0 | 205 |
| 64 | 0 | 0 |
| 0 | 222 | 178 |
| 0 | 178 | 19 |
| 64 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 240 | 0 |
G:=sub<GL(3,GF(241))| [1,0,0,0,205,0,0,0,205],[64,0,0,0,222,178,0,178,19],[64,0,0,0,0,240,0,1,0] >;
C5×C16⋊5C4 in GAP, Magma, Sage, TeX
C_5\times C_{16}\rtimes_5C_4 % in TeX
G:=Group("C5xC16:5C4"); // GroupNames label
G:=SmallGroup(320,151);
// by ID
G=gap.SmallGroup(320,151);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,2269,288,136,124]);
// Polycyclic
G:=Group<a,b,c|a^5=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
Export