direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C16⋊4C4, C16⋊4C20, C80⋊18C4, C40.19Q8, C20.19Q16, C10.11SD32, C8.3(C5×Q8), C4.2(C5×Q16), (C2×C16).6C10, C8.14(C2×C20), (C2×C80).16C2, (C2×C10).52D8, C20.88(C4⋊C4), C2.D8.3C10, C2.3(C5×SD32), C40.123(C2×C4), (C2×C20).409D4, C22.11(C5×D8), C10.21(C2.D8), (C2×C40).419C22, C4.8(C5×C4⋊C4), C2.4(C5×C2.D8), (C2×C4).63(C5×D4), (C2×C8).74(C2×C10), (C5×C2.D8).10C2, SmallGroup(320,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C16⋊4C4
G = < a,b,c | a5=b16=c4=1, ab=ba, ac=ca, cbc-1=b7 >
Smallest permutation representation of C5×C16⋊4C4
►Regular action on 320 points
Generators in S320
(1 214 180 66 194)(2 215 181 67 195)(3 216 182 68 196)(4 217 183 69 197)(5 218 184 70 198)(6 219 185 71 199)(7 220 186 72 200)(8 221 187 73 201)(9 222 188 74 202)(10 223 189 75 203)(11 224 190 76 204)(12 209 191 77 205)(13 210 192 78 206)(14 211 177 79 207)(15 212 178 80 208)(16 213 179 65 193)(17 277 251 309 228)(18 278 252 310 229)(19 279 253 311 230)(20 280 254 312 231)(21 281 255 313 232)(22 282 256 314 233)(23 283 241 315 234)(24 284 242 316 235)(25 285 243 317 236)(26 286 244 318 237)(27 287 245 319 238)(28 288 246 320 239)(29 273 247 305 240)(30 274 248 306 225)(31 275 249 307 226)(32 276 250 308 227)(33 258 139 109 173)(34 259 140 110 174)(35 260 141 111 175)(36 261 142 112 176)(37 262 143 97 161)(38 263 144 98 162)(39 264 129 99 163)(40 265 130 100 164)(41 266 131 101 165)(42 267 132 102 166)(43 268 133 103 167)(44 269 134 104 168)(45 270 135 105 169)(46 271 136 106 170)(47 272 137 107 171)(48 257 138 108 172)(49 145 117 295 82)(50 146 118 296 83)(51 147 119 297 84)(52 148 120 298 85)(53 149 121 299 86)(54 150 122 300 87)(55 151 123 301 88)(56 152 124 302 89)(57 153 125 303 90)(58 154 126 304 91)(59 155 127 289 92)(60 156 128 290 93)(61 157 113 291 94)(62 158 114 292 95)(63 159 115 293 96)(64 160 116 294 81)
(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)
(1 318 301 132)(2 309 302 139)(3 316 303 130)(4 307 304 137)(5 314 289 144)(6 305 290 135)(7 312 291 142)(8 319 292 133)(9 310 293 140)(10 317 294 131)(11 308 295 138)(12 315 296 129)(13 306 297 136)(14 313 298 143)(15 320 299 134)(16 311 300 141)(17 56 173 181)(18 63 174 188)(19 54 175 179)(20 61 176 186)(21 52 161 177)(22 59 162 184)(23 50 163 191)(24 57 164 182)(25 64 165 189)(26 55 166 180)(27 62 167 187)(28 53 168 178)(29 60 169 185)(30 51 170 192)(31 58 171 183)(32 49 172 190)(33 67 277 152)(34 74 278 159)(35 65 279 150)(36 72 280 157)(37 79 281 148)(38 70 282 155)(39 77 283 146)(40 68 284 153)(41 75 285 160)(42 66 286 151)(43 73 287 158)(44 80 288 149)(45 71 273 156)(46 78 274 147)(47 69 275 154)(48 76 276 145)(81 101 223 236)(82 108 224 227)(83 99 209 234)(84 106 210 225)(85 97 211 232)(86 104 212 239)(87 111 213 230)(88 102 214 237)(89 109 215 228)(90 100 216 235)(91 107 217 226)(92 98 218 233)(93 105 219 240)(94 112 220 231)(95 103 221 238)(96 110 222 229)(113 261 200 254)(114 268 201 245)(115 259 202 252)(116 266 203 243)(117 257 204 250)(118 264 205 241)(119 271 206 248)(120 262 207 255)(121 269 208 246)(122 260 193 253)(123 267 194 244)(124 258 195 251)(125 265 196 242)(126 272 197 249)(127 263 198 256)(128 270 199 247)
G:=sub<Sym(320)| (1,214,180,66,194)(2,215,181,67,195)(3,216,182,68,196)(4,217,183,69,197)(5,218,184,70,198)(6,219,185,71,199)(7,220,186,72,200)(8,221,187,73,201)(9,222,188,74,202)(10,223,189,75,203)(11,224,190,76,204)(12,209,191,77,205)(13,210,192,78,206)(14,211,177,79,207)(15,212,178,80,208)(16,213,179,65,193)(17,277,251,309,228)(18,278,252,310,229)(19,279,253,311,230)(20,280,254,312,231)(21,281,255,313,232)(22,282,256,314,233)(23,283,241,315,234)(24,284,242,316,235)(25,285,243,317,236)(26,286,244,318,237)(27,287,245,319,238)(28,288,246,320,239)(29,273,247,305,240)(30,274,248,306,225)(31,275,249,307,226)(32,276,250,308,227)(33,258,139,109,173)(34,259,140,110,174)(35,260,141,111,175)(36,261,142,112,176)(37,262,143,97,161)(38,263,144,98,162)(39,264,129,99,163)(40,265,130,100,164)(41,266,131,101,165)(42,267,132,102,166)(43,268,133,103,167)(44,269,134,104,168)(45,270,135,105,169)(46,271,136,106,170)(47,272,137,107,171)(48,257,138,108,172)(49,145,117,295,82)(50,146,118,296,83)(51,147,119,297,84)(52,148,120,298,85)(53,149,121,299,86)(54,150,122,300,87)(55,151,123,301,88)(56,152,124,302,89)(57,153,125,303,90)(58,154,126,304,91)(59,155,127,289,92)(60,156,128,290,93)(61,157,113,291,94)(62,158,114,292,95)(63,159,115,293,96)(64,160,116,294,81), (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), (1,318,301,132)(2,309,302,139)(3,316,303,130)(4,307,304,137)(5,314,289,144)(6,305,290,135)(7,312,291,142)(8,319,292,133)(9,310,293,140)(10,317,294,131)(11,308,295,138)(12,315,296,129)(13,306,297,136)(14,313,298,143)(15,320,299,134)(16,311,300,141)(17,56,173,181)(18,63,174,188)(19,54,175,179)(20,61,176,186)(21,52,161,177)(22,59,162,184)(23,50,163,191)(24,57,164,182)(25,64,165,189)(26,55,166,180)(27,62,167,187)(28,53,168,178)(29,60,169,185)(30,51,170,192)(31,58,171,183)(32,49,172,190)(33,67,277,152)(34,74,278,159)(35,65,279,150)(36,72,280,157)(37,79,281,148)(38,70,282,155)(39,77,283,146)(40,68,284,153)(41,75,285,160)(42,66,286,151)(43,73,287,158)(44,80,288,149)(45,71,273,156)(46,78,274,147)(47,69,275,154)(48,76,276,145)(81,101,223,236)(82,108,224,227)(83,99,209,234)(84,106,210,225)(85,97,211,232)(86,104,212,239)(87,111,213,230)(88,102,214,237)(89,109,215,228)(90,100,216,235)(91,107,217,226)(92,98,218,233)(93,105,219,240)(94,112,220,231)(95,103,221,238)(96,110,222,229)(113,261,200,254)(114,268,201,245)(115,259,202,252)(116,266,203,243)(117,257,204,250)(118,264,205,241)(119,271,206,248)(120,262,207,255)(121,269,208,246)(122,260,193,253)(123,267,194,244)(124,258,195,251)(125,265,196,242)(126,272,197,249)(127,263,198,256)(128,270,199,247)>;
G:=Group( (1,214,180,66,194)(2,215,181,67,195)(3,216,182,68,196)(4,217,183,69,197)(5,218,184,70,198)(6,219,185,71,199)(7,220,186,72,200)(8,221,187,73,201)(9,222,188,74,202)(10,223,189,75,203)(11,224,190,76,204)(12,209,191,77,205)(13,210,192,78,206)(14,211,177,79,207)(15,212,178,80,208)(16,213,179,65,193)(17,277,251,309,228)(18,278,252,310,229)(19,279,253,311,230)(20,280,254,312,231)(21,281,255,313,232)(22,282,256,314,233)(23,283,241,315,234)(24,284,242,316,235)(25,285,243,317,236)(26,286,244,318,237)(27,287,245,319,238)(28,288,246,320,239)(29,273,247,305,240)(30,274,248,306,225)(31,275,249,307,226)(32,276,250,308,227)(33,258,139,109,173)(34,259,140,110,174)(35,260,141,111,175)(36,261,142,112,176)(37,262,143,97,161)(38,263,144,98,162)(39,264,129,99,163)(40,265,130,100,164)(41,266,131,101,165)(42,267,132,102,166)(43,268,133,103,167)(44,269,134,104,168)(45,270,135,105,169)(46,271,136,106,170)(47,272,137,107,171)(48,257,138,108,172)(49,145,117,295,82)(50,146,118,296,83)(51,147,119,297,84)(52,148,120,298,85)(53,149,121,299,86)(54,150,122,300,87)(55,151,123,301,88)(56,152,124,302,89)(57,153,125,303,90)(58,154,126,304,91)(59,155,127,289,92)(60,156,128,290,93)(61,157,113,291,94)(62,158,114,292,95)(63,159,115,293,96)(64,160,116,294,81), (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), (1,318,301,132)(2,309,302,139)(3,316,303,130)(4,307,304,137)(5,314,289,144)(6,305,290,135)(7,312,291,142)(8,319,292,133)(9,310,293,140)(10,317,294,131)(11,308,295,138)(12,315,296,129)(13,306,297,136)(14,313,298,143)(15,320,299,134)(16,311,300,141)(17,56,173,181)(18,63,174,188)(19,54,175,179)(20,61,176,186)(21,52,161,177)(22,59,162,184)(23,50,163,191)(24,57,164,182)(25,64,165,189)(26,55,166,180)(27,62,167,187)(28,53,168,178)(29,60,169,185)(30,51,170,192)(31,58,171,183)(32,49,172,190)(33,67,277,152)(34,74,278,159)(35,65,279,150)(36,72,280,157)(37,79,281,148)(38,70,282,155)(39,77,283,146)(40,68,284,153)(41,75,285,160)(42,66,286,151)(43,73,287,158)(44,80,288,149)(45,71,273,156)(46,78,274,147)(47,69,275,154)(48,76,276,145)(81,101,223,236)(82,108,224,227)(83,99,209,234)(84,106,210,225)(85,97,211,232)(86,104,212,239)(87,111,213,230)(88,102,214,237)(89,109,215,228)(90,100,216,235)(91,107,217,226)(92,98,218,233)(93,105,219,240)(94,112,220,231)(95,103,221,238)(96,110,222,229)(113,261,200,254)(114,268,201,245)(115,259,202,252)(116,266,203,243)(117,257,204,250)(118,264,205,241)(119,271,206,248)(120,262,207,255)(121,269,208,246)(122,260,193,253)(123,267,194,244)(124,258,195,251)(125,265,196,242)(126,272,197,249)(127,263,198,256)(128,270,199,247) );
G=PermutationGroup([[(1,214,180,66,194),(2,215,181,67,195),(3,216,182,68,196),(4,217,183,69,197),(5,218,184,70,198),(6,219,185,71,199),(7,220,186,72,200),(8,221,187,73,201),(9,222,188,74,202),(10,223,189,75,203),(11,224,190,76,204),(12,209,191,77,205),(13,210,192,78,206),(14,211,177,79,207),(15,212,178,80,208),(16,213,179,65,193),(17,277,251,309,228),(18,278,252,310,229),(19,279,253,311,230),(20,280,254,312,231),(21,281,255,313,232),(22,282,256,314,233),(23,283,241,315,234),(24,284,242,316,235),(25,285,243,317,236),(26,286,244,318,237),(27,287,245,319,238),(28,288,246,320,239),(29,273,247,305,240),(30,274,248,306,225),(31,275,249,307,226),(32,276,250,308,227),(33,258,139,109,173),(34,259,140,110,174),(35,260,141,111,175),(36,261,142,112,176),(37,262,143,97,161),(38,263,144,98,162),(39,264,129,99,163),(40,265,130,100,164),(41,266,131,101,165),(42,267,132,102,166),(43,268,133,103,167),(44,269,134,104,168),(45,270,135,105,169),(46,271,136,106,170),(47,272,137,107,171),(48,257,138,108,172),(49,145,117,295,82),(50,146,118,296,83),(51,147,119,297,84),(52,148,120,298,85),(53,149,121,299,86),(54,150,122,300,87),(55,151,123,301,88),(56,152,124,302,89),(57,153,125,303,90),(58,154,126,304,91),(59,155,127,289,92),(60,156,128,290,93),(61,157,113,291,94),(62,158,114,292,95),(63,159,115,293,96),(64,160,116,294,81)], [(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)], [(1,318,301,132),(2,309,302,139),(3,316,303,130),(4,307,304,137),(5,314,289,144),(6,305,290,135),(7,312,291,142),(8,319,292,133),(9,310,293,140),(10,317,294,131),(11,308,295,138),(12,315,296,129),(13,306,297,136),(14,313,298,143),(15,320,299,134),(16,311,300,141),(17,56,173,181),(18,63,174,188),(19,54,175,179),(20,61,176,186),(21,52,161,177),(22,59,162,184),(23,50,163,191),(24,57,164,182),(25,64,165,189),(26,55,166,180),(27,62,167,187),(28,53,168,178),(29,60,169,185),(30,51,170,192),(31,58,171,183),(32,49,172,190),(33,67,277,152),(34,74,278,159),(35,65,279,150),(36,72,280,157),(37,79,281,148),(38,70,282,155),(39,77,283,146),(40,68,284,153),(41,75,285,160),(42,66,286,151),(43,73,287,158),(44,80,288,149),(45,71,273,156),(46,78,274,147),(47,69,275,154),(48,76,276,145),(81,101,223,236),(82,108,224,227),(83,99,209,234),(84,106,210,225),(85,97,211,232),(86,104,212,239),(87,111,213,230),(88,102,214,237),(89,109,215,228),(90,100,216,235),(91,107,217,226),(92,98,218,233),(93,105,219,240),(94,112,220,231),(95,103,221,238),(96,110,222,229),(113,261,200,254),(114,268,201,245),(115,259,202,252),(116,266,203,243),(117,257,204,250),(118,264,205,241),(119,271,206,248),(120,262,207,255),(121,269,208,246),(122,260,193,253),(123,267,194,244),(124,258,195,251),(125,265,196,242),(126,272,197,249),(127,263,198,256),(128,270,199,247)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 16A | ··· | 16H | 20A | ··· | 20H | 20I | ··· | 20X | 40A | ··· | 40P | 80A | ··· | 80AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | Q8 | D4 | Q16 | D8 | SD32 | C5×Q8 | C5×D4 | C5×Q16 | C5×D8 | C5×SD32 |
| kernel | C5×C16⋊4C4 | C5×C2.D8 | C2×C80 | C80 | C16⋊4C4 | C2.D8 | C2×C16 | C16 | C40 | C2×C20 | C20 | C2×C10 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 1 | 2 | 2 | 8 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C5×C16⋊4C4 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 87 | 0 |
| 0 | 0 | 87 |
| 1 | 0 | 0 |
| 0 | 138 | 200 |
| 0 | 41 | 138 |
| 177 | 0 | 0 |
| 0 | 232 | 20 |
| 0 | 20 | 9 |
G:=sub<GL(3,GF(241))| [1,0,0,0,87,0,0,0,87],[1,0,0,0,138,41,0,200,138],[177,0,0,0,232,20,0,20,9] >;
C5×C16⋊4C4 in GAP, Magma, Sage, TeX
C_5\times C_{16}\rtimes_4C_4 % in TeX
G:=Group("C5xC16:4C4"); // GroupNames label
G:=SmallGroup(320,172);
// by ID
G=gap.SmallGroup(320,172);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1828,2803,360,10085,124]);
// 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^7>;
// generators/relations
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