direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C5⋊C16, C20⋊2C16, C42.16F5, C20.17C42, C5⋊2(C4×C16), C5⋊2C8.7C8, (C2×C20).7C8, C10.5(C4×C8), C4.15(C4×F5), (C4×C20).14C4, C20.18(C2×C8), C10.6(C2×C16), C4.13(D5⋊C8), C2.1(C4×C5⋊C8), C2.1(C2×C5⋊C16), (C2×C5⋊C16).5C2, (C2×C4).7(C5⋊C8), C22.7(C2×C5⋊C8), (C4×C5⋊2C8).20C2, (C2×C5⋊2C8).29C4, (C2×C10).23(C2×C8), C5⋊2C8.34(C2×C4), (C2×C4).147(C2×F5), (C2×C20).153(C2×C4), (C2×C5⋊2C8).340C22, SmallGroup(320,195)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×C5⋊C16 |
Generators and relations for C4×C5⋊C16
G = < a,b,c | a4=b5=c16=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C4×C5⋊C16
►Regular action on 320 points
Generators in S320
(1 128 55 212)(2 113 56 213)(3 114 57 214)(4 115 58 215)(5 116 59 216)(6 117 60 217)(7 118 61 218)(8 119 62 219)(9 120 63 220)(10 121 64 221)(11 122 49 222)(12 123 50 223)(13 124 51 224)(14 125 52 209)(15 126 53 210)(16 127 54 211)(17 257 106 242)(18 258 107 243)(19 259 108 244)(20 260 109 245)(21 261 110 246)(22 262 111 247)(23 263 112 248)(24 264 97 249)(25 265 98 250)(26 266 99 251)(27 267 100 252)(28 268 101 253)(29 269 102 254)(30 270 103 255)(31 271 104 256)(32 272 105 241)(33 145 92 203)(34 146 93 204)(35 147 94 205)(36 148 95 206)(37 149 96 207)(38 150 81 208)(39 151 82 193)(40 152 83 194)(41 153 84 195)(42 154 85 196)(43 155 86 197)(44 156 87 198)(45 157 88 199)(46 158 89 200)(47 159 90 201)(48 160 91 202)(65 287 179 130)(66 288 180 131)(67 273 181 132)(68 274 182 133)(69 275 183 134)(70 276 184 135)(71 277 185 136)(72 278 186 137)(73 279 187 138)(74 280 188 139)(75 281 189 140)(76 282 190 141)(77 283 191 142)(78 284 192 143)(79 285 177 144)(80 286 178 129)(161 313 300 226)(162 314 301 227)(163 315 302 228)(164 316 303 229)(165 317 304 230)(166 318 289 231)(167 319 290 232)(168 320 291 233)(169 305 292 234)(170 306 293 235)(171 307 294 236)(172 308 295 237)(173 309 296 238)(174 310 297 239)(175 311 298 240)(176 312 299 225)
(1 27 228 95 177)(2 96 28 178 229)(3 179 81 230 29)(4 231 180 30 82)(5 31 232 83 181)(6 84 32 182 233)(7 183 85 234 17)(8 235 184 18 86)(9 19 236 87 185)(10 88 20 186 237)(11 187 89 238 21)(12 239 188 22 90)(13 23 240 91 189)(14 92 24 190 225)(15 191 93 226 25)(16 227 192 26 94)(33 97 76 312 52)(34 313 98 53 77)(35 54 314 78 99)(36 79 55 100 315)(37 101 80 316 56)(38 317 102 57 65)(39 58 318 66 103)(40 67 59 104 319)(41 105 68 320 60)(42 305 106 61 69)(43 62 306 70 107)(44 71 63 108 307)(45 109 72 308 64)(46 309 110 49 73)(47 50 310 74 111)(48 75 51 112 311)(113 207 268 129 164)(114 130 208 165 269)(115 166 131 270 193)(116 271 167 194 132)(117 195 272 133 168)(118 134 196 169 257)(119 170 135 258 197)(120 259 171 198 136)(121 199 260 137 172)(122 138 200 173 261)(123 174 139 262 201)(124 263 175 202 140)(125 203 264 141 176)(126 142 204 161 265)(127 162 143 266 205)(128 267 163 206 144)(145 249 282 299 209)(146 300 250 210 283)(147 211 301 284 251)(148 285 212 252 302)(149 253 286 303 213)(150 304 254 214 287)(151 215 289 288 255)(152 273 216 256 290)(153 241 274 291 217)(154 292 242 218 275)(155 219 293 276 243)(156 277 220 244 294)(157 245 278 295 221)(158 296 246 222 279)(159 223 297 280 247)(160 281 224 248 298)
(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)| (1,128,55,212)(2,113,56,213)(3,114,57,214)(4,115,58,215)(5,116,59,216)(6,117,60,217)(7,118,61,218)(8,119,62,219)(9,120,63,220)(10,121,64,221)(11,122,49,222)(12,123,50,223)(13,124,51,224)(14,125,52,209)(15,126,53,210)(16,127,54,211)(17,257,106,242)(18,258,107,243)(19,259,108,244)(20,260,109,245)(21,261,110,246)(22,262,111,247)(23,263,112,248)(24,264,97,249)(25,265,98,250)(26,266,99,251)(27,267,100,252)(28,268,101,253)(29,269,102,254)(30,270,103,255)(31,271,104,256)(32,272,105,241)(33,145,92,203)(34,146,93,204)(35,147,94,205)(36,148,95,206)(37,149,96,207)(38,150,81,208)(39,151,82,193)(40,152,83,194)(41,153,84,195)(42,154,85,196)(43,155,86,197)(44,156,87,198)(45,157,88,199)(46,158,89,200)(47,159,90,201)(48,160,91,202)(65,287,179,130)(66,288,180,131)(67,273,181,132)(68,274,182,133)(69,275,183,134)(70,276,184,135)(71,277,185,136)(72,278,186,137)(73,279,187,138)(74,280,188,139)(75,281,189,140)(76,282,190,141)(77,283,191,142)(78,284,192,143)(79,285,177,144)(80,286,178,129)(161,313,300,226)(162,314,301,227)(163,315,302,228)(164,316,303,229)(165,317,304,230)(166,318,289,231)(167,319,290,232)(168,320,291,233)(169,305,292,234)(170,306,293,235)(171,307,294,236)(172,308,295,237)(173,309,296,238)(174,310,297,239)(175,311,298,240)(176,312,299,225), (1,27,228,95,177)(2,96,28,178,229)(3,179,81,230,29)(4,231,180,30,82)(5,31,232,83,181)(6,84,32,182,233)(7,183,85,234,17)(8,235,184,18,86)(9,19,236,87,185)(10,88,20,186,237)(11,187,89,238,21)(12,239,188,22,90)(13,23,240,91,189)(14,92,24,190,225)(15,191,93,226,25)(16,227,192,26,94)(33,97,76,312,52)(34,313,98,53,77)(35,54,314,78,99)(36,79,55,100,315)(37,101,80,316,56)(38,317,102,57,65)(39,58,318,66,103)(40,67,59,104,319)(41,105,68,320,60)(42,305,106,61,69)(43,62,306,70,107)(44,71,63,108,307)(45,109,72,308,64)(46,309,110,49,73)(47,50,310,74,111)(48,75,51,112,311)(113,207,268,129,164)(114,130,208,165,269)(115,166,131,270,193)(116,271,167,194,132)(117,195,272,133,168)(118,134,196,169,257)(119,170,135,258,197)(120,259,171,198,136)(121,199,260,137,172)(122,138,200,173,261)(123,174,139,262,201)(124,263,175,202,140)(125,203,264,141,176)(126,142,204,161,265)(127,162,143,266,205)(128,267,163,206,144)(145,249,282,299,209)(146,300,250,210,283)(147,211,301,284,251)(148,285,212,252,302)(149,253,286,303,213)(150,304,254,214,287)(151,215,289,288,255)(152,273,216,256,290)(153,241,274,291,217)(154,292,242,218,275)(155,219,293,276,243)(156,277,220,244,294)(157,245,278,295,221)(158,296,246,222,279)(159,223,297,280,247)(160,281,224,248,298), (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( (1,128,55,212)(2,113,56,213)(3,114,57,214)(4,115,58,215)(5,116,59,216)(6,117,60,217)(7,118,61,218)(8,119,62,219)(9,120,63,220)(10,121,64,221)(11,122,49,222)(12,123,50,223)(13,124,51,224)(14,125,52,209)(15,126,53,210)(16,127,54,211)(17,257,106,242)(18,258,107,243)(19,259,108,244)(20,260,109,245)(21,261,110,246)(22,262,111,247)(23,263,112,248)(24,264,97,249)(25,265,98,250)(26,266,99,251)(27,267,100,252)(28,268,101,253)(29,269,102,254)(30,270,103,255)(31,271,104,256)(32,272,105,241)(33,145,92,203)(34,146,93,204)(35,147,94,205)(36,148,95,206)(37,149,96,207)(38,150,81,208)(39,151,82,193)(40,152,83,194)(41,153,84,195)(42,154,85,196)(43,155,86,197)(44,156,87,198)(45,157,88,199)(46,158,89,200)(47,159,90,201)(48,160,91,202)(65,287,179,130)(66,288,180,131)(67,273,181,132)(68,274,182,133)(69,275,183,134)(70,276,184,135)(71,277,185,136)(72,278,186,137)(73,279,187,138)(74,280,188,139)(75,281,189,140)(76,282,190,141)(77,283,191,142)(78,284,192,143)(79,285,177,144)(80,286,178,129)(161,313,300,226)(162,314,301,227)(163,315,302,228)(164,316,303,229)(165,317,304,230)(166,318,289,231)(167,319,290,232)(168,320,291,233)(169,305,292,234)(170,306,293,235)(171,307,294,236)(172,308,295,237)(173,309,296,238)(174,310,297,239)(175,311,298,240)(176,312,299,225), (1,27,228,95,177)(2,96,28,178,229)(3,179,81,230,29)(4,231,180,30,82)(5,31,232,83,181)(6,84,32,182,233)(7,183,85,234,17)(8,235,184,18,86)(9,19,236,87,185)(10,88,20,186,237)(11,187,89,238,21)(12,239,188,22,90)(13,23,240,91,189)(14,92,24,190,225)(15,191,93,226,25)(16,227,192,26,94)(33,97,76,312,52)(34,313,98,53,77)(35,54,314,78,99)(36,79,55,100,315)(37,101,80,316,56)(38,317,102,57,65)(39,58,318,66,103)(40,67,59,104,319)(41,105,68,320,60)(42,305,106,61,69)(43,62,306,70,107)(44,71,63,108,307)(45,109,72,308,64)(46,309,110,49,73)(47,50,310,74,111)(48,75,51,112,311)(113,207,268,129,164)(114,130,208,165,269)(115,166,131,270,193)(116,271,167,194,132)(117,195,272,133,168)(118,134,196,169,257)(119,170,135,258,197)(120,259,171,198,136)(121,199,260,137,172)(122,138,200,173,261)(123,174,139,262,201)(124,263,175,202,140)(125,203,264,141,176)(126,142,204,161,265)(127,162,143,266,205)(128,267,163,206,144)(145,249,282,299,209)(146,300,250,210,283)(147,211,301,284,251)(148,285,212,252,302)(149,253,286,303,213)(150,304,254,214,287)(151,215,289,288,255)(152,273,216,256,290)(153,241,274,291,217)(154,292,242,218,275)(155,219,293,276,243)(156,277,220,244,294)(157,245,278,295,221)(158,296,246,222,279)(159,223,297,280,247)(160,281,224,248,298), (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([[(1,128,55,212),(2,113,56,213),(3,114,57,214),(4,115,58,215),(5,116,59,216),(6,117,60,217),(7,118,61,218),(8,119,62,219),(9,120,63,220),(10,121,64,221),(11,122,49,222),(12,123,50,223),(13,124,51,224),(14,125,52,209),(15,126,53,210),(16,127,54,211),(17,257,106,242),(18,258,107,243),(19,259,108,244),(20,260,109,245),(21,261,110,246),(22,262,111,247),(23,263,112,248),(24,264,97,249),(25,265,98,250),(26,266,99,251),(27,267,100,252),(28,268,101,253),(29,269,102,254),(30,270,103,255),(31,271,104,256),(32,272,105,241),(33,145,92,203),(34,146,93,204),(35,147,94,205),(36,148,95,206),(37,149,96,207),(38,150,81,208),(39,151,82,193),(40,152,83,194),(41,153,84,195),(42,154,85,196),(43,155,86,197),(44,156,87,198),(45,157,88,199),(46,158,89,200),(47,159,90,201),(48,160,91,202),(65,287,179,130),(66,288,180,131),(67,273,181,132),(68,274,182,133),(69,275,183,134),(70,276,184,135),(71,277,185,136),(72,278,186,137),(73,279,187,138),(74,280,188,139),(75,281,189,140),(76,282,190,141),(77,283,191,142),(78,284,192,143),(79,285,177,144),(80,286,178,129),(161,313,300,226),(162,314,301,227),(163,315,302,228),(164,316,303,229),(165,317,304,230),(166,318,289,231),(167,319,290,232),(168,320,291,233),(169,305,292,234),(170,306,293,235),(171,307,294,236),(172,308,295,237),(173,309,296,238),(174,310,297,239),(175,311,298,240),(176,312,299,225)], [(1,27,228,95,177),(2,96,28,178,229),(3,179,81,230,29),(4,231,180,30,82),(5,31,232,83,181),(6,84,32,182,233),(7,183,85,234,17),(8,235,184,18,86),(9,19,236,87,185),(10,88,20,186,237),(11,187,89,238,21),(12,239,188,22,90),(13,23,240,91,189),(14,92,24,190,225),(15,191,93,226,25),(16,227,192,26,94),(33,97,76,312,52),(34,313,98,53,77),(35,54,314,78,99),(36,79,55,100,315),(37,101,80,316,56),(38,317,102,57,65),(39,58,318,66,103),(40,67,59,104,319),(41,105,68,320,60),(42,305,106,61,69),(43,62,306,70,107),(44,71,63,108,307),(45,109,72,308,64),(46,309,110,49,73),(47,50,310,74,111),(48,75,51,112,311),(113,207,268,129,164),(114,130,208,165,269),(115,166,131,270,193),(116,271,167,194,132),(117,195,272,133,168),(118,134,196,169,257),(119,170,135,258,197),(120,259,171,198,136),(121,199,260,137,172),(122,138,200,173,261),(123,174,139,262,201),(124,263,175,202,140),(125,203,264,141,176),(126,142,204,161,265),(127,162,143,266,205),(128,267,163,206,144),(145,249,282,299,209),(146,300,250,210,283),(147,211,301,284,251),(148,285,212,252,302),(149,253,286,303,213),(150,304,254,214,287),(151,215,289,288,255),(152,273,216,256,290),(153,241,274,291,217),(154,292,242,218,275),(155,219,293,276,243),(156,277,220,244,294),(157,245,278,295,221),(158,296,246,222,279),(159,223,297,280,247),(160,281,224,248,298)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5 | 8A | ··· | 8P | 10A | 10B | 10C | 16A | ··· | 16AF | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | 5 | ··· | 5 | 4 | 4 | 4 | 5 | ··· | 5 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 | F5 | C5⋊C8 | C2×F5 | C5⋊C16 | D5⋊C8 | C4×F5 |
| kernel | C4×C5⋊C16 | C4×C5⋊2C8 | C2×C5⋊C16 | C5⋊C16 | C2×C5⋊2C8 | C4×C20 | C5⋊2C8 | C2×C20 | C20 | C42 | C2×C4 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 32 | 1 | 2 | 1 | 8 | 2 | 2 |
Matrix representation of C4×C5⋊C16 ►in GL5(𝔽241)
| 177 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 |
| 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 1 | 240 |
| 177 | 0 | 0 | 0 | 0 |
| 0 | 230 | 2 | 76 | 88 |
| 0 | 65 | 90 | 162 | 77 |
| 0 | 151 | 79 | 164 | 153 |
| 0 | 153 | 155 | 11 | 239 |
G:=sub<GL(5,GF(241))| [177,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,240,240,240,240],[177,0,0,0,0,0,230,65,151,153,0,2,90,79,155,0,76,162,164,11,0,88,77,153,239] >;
C4×C5⋊C16 in GAP, Magma, Sage, TeX
C_4\times C_5\rtimes C_{16} % in TeX
G:=Group("C4xC5:C16"); // GroupNames label
G:=SmallGroup(320,195);
// by ID
G=gap.SmallGroup(320,195);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,100,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^4=b^5=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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