direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C5⋊2C16, C20⋊4C16, C40.9C8, C20.43C42, C42.13Dic5, C5⋊4(C4×C16), C8.38(C4×D5), (C4×C8).18D5, C8.5(C5⋊2C8), (C4×C20).34C4, (C2×C20).19C8, C20.74(C2×C8), (C2×C40).46C4, C10.14(C4×C8), (C4×C40).18C2, C40.96(C2×C4), C10.16(C2×C16), (C2×C8).329D10, (C2×C8).17Dic5, C4.13(C4×Dic5), (C2×C40).394C22, C2.2(C4×C5⋊2C8), C2.1(C2×C5⋊2C16), C4.12(C2×C5⋊2C8), (C2×C10).53(C2×C8), (C2×C4).7(C5⋊2C8), C22.7(C2×C5⋊2C8), (C2×C5⋊2C16).13C2, (C2×C20).482(C2×C4), (C2×C4).90(C2×Dic5), SmallGroup(320,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C4×C5⋊2C16 |
Generators and relations for C4×C5⋊2C16
G = < a,b,c | a4=b5=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×C5⋊2C16
►Regular action on 320 points
Generators in S320
(1 167 115 298)(2 168 116 299)(3 169 117 300)(4 170 118 301)(5 171 119 302)(6 172 120 303)(7 173 121 304)(8 174 122 289)(9 175 123 290)(10 176 124 291)(11 161 125 292)(12 162 126 293)(13 163 127 294)(14 164 128 295)(15 165 113 296)(16 166 114 297)(17 201 154 93)(18 202 155 94)(19 203 156 95)(20 204 157 96)(21 205 158 81)(22 206 159 82)(23 207 160 83)(24 208 145 84)(25 193 146 85)(26 194 147 86)(27 195 148 87)(28 196 149 88)(29 197 150 89)(30 198 151 90)(31 199 152 91)(32 200 153 92)(33 319 238 67)(34 320 239 68)(35 305 240 69)(36 306 225 70)(37 307 226 71)(38 308 227 72)(39 309 228 73)(40 310 229 74)(41 311 230 75)(42 312 231 76)(43 313 232 77)(44 314 233 78)(45 315 234 79)(46 316 235 80)(47 317 236 65)(48 318 237 66)(49 272 250 139)(50 257 251 140)(51 258 252 141)(52 259 253 142)(53 260 254 143)(54 261 255 144)(55 262 256 129)(56 263 241 130)(57 264 242 131)(58 265 243 132)(59 266 244 133)(60 267 245 134)(61 268 246 135)(62 269 247 136)(63 270 248 137)(64 271 249 138)(97 216 274 178)(98 217 275 179)(99 218 276 180)(100 219 277 181)(101 220 278 182)(102 221 279 183)(103 222 280 184)(104 223 281 185)(105 224 282 186)(106 209 283 187)(107 210 284 188)(108 211 285 189)(109 212 286 190)(110 213 287 191)(111 214 288 192)(112 215 273 177)
(1 51 155 38 209)(2 210 39 156 52)(3 53 157 40 211)(4 212 41 158 54)(5 55 159 42 213)(6 214 43 160 56)(7 57 145 44 215)(8 216 45 146 58)(9 59 147 46 217)(10 218 47 148 60)(11 61 149 48 219)(12 220 33 150 62)(13 63 151 34 221)(14 222 35 152 64)(15 49 153 36 223)(16 224 37 154 50)(17 251 114 186 226)(18 227 187 115 252)(19 253 116 188 228)(20 229 189 117 254)(21 255 118 190 230)(22 231 191 119 256)(23 241 120 192 232)(24 233 177 121 242)(25 243 122 178 234)(26 235 179 123 244)(27 245 124 180 236)(28 237 181 125 246)(29 247 126 182 238)(30 239 183 127 248)(31 249 128 184 240)(32 225 185 113 250)(65 195 134 291 99)(66 100 292 135 196)(67 197 136 293 101)(68 102 294 137 198)(69 199 138 295 103)(70 104 296 139 200)(71 201 140 297 105)(72 106 298 141 202)(73 203 142 299 107)(74 108 300 143 204)(75 205 144 301 109)(76 110 302 129 206)(77 207 130 303 111)(78 112 304 131 208)(79 193 132 289 97)(80 98 290 133 194)(81 261 170 286 311)(82 312 287 171 262)(83 263 172 288 313)(84 314 273 173 264)(85 265 174 274 315)(86 316 275 175 266)(87 267 176 276 317)(88 318 277 161 268)(89 269 162 278 319)(90 320 279 163 270)(91 271 164 280 305)(92 306 281 165 272)(93 257 166 282 307)(94 308 283 167 258)(95 259 168 284 309)(96 310 285 169 260)
(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)| (1,167,115,298)(2,168,116,299)(3,169,117,300)(4,170,118,301)(5,171,119,302)(6,172,120,303)(7,173,121,304)(8,174,122,289)(9,175,123,290)(10,176,124,291)(11,161,125,292)(12,162,126,293)(13,163,127,294)(14,164,128,295)(15,165,113,296)(16,166,114,297)(17,201,154,93)(18,202,155,94)(19,203,156,95)(20,204,157,96)(21,205,158,81)(22,206,159,82)(23,207,160,83)(24,208,145,84)(25,193,146,85)(26,194,147,86)(27,195,148,87)(28,196,149,88)(29,197,150,89)(30,198,151,90)(31,199,152,91)(32,200,153,92)(33,319,238,67)(34,320,239,68)(35,305,240,69)(36,306,225,70)(37,307,226,71)(38,308,227,72)(39,309,228,73)(40,310,229,74)(41,311,230,75)(42,312,231,76)(43,313,232,77)(44,314,233,78)(45,315,234,79)(46,316,235,80)(47,317,236,65)(48,318,237,66)(49,272,250,139)(50,257,251,140)(51,258,252,141)(52,259,253,142)(53,260,254,143)(54,261,255,144)(55,262,256,129)(56,263,241,130)(57,264,242,131)(58,265,243,132)(59,266,244,133)(60,267,245,134)(61,268,246,135)(62,269,247,136)(63,270,248,137)(64,271,249,138)(97,216,274,178)(98,217,275,179)(99,218,276,180)(100,219,277,181)(101,220,278,182)(102,221,279,183)(103,222,280,184)(104,223,281,185)(105,224,282,186)(106,209,283,187)(107,210,284,188)(108,211,285,189)(109,212,286,190)(110,213,287,191)(111,214,288,192)(112,215,273,177), (1,51,155,38,209)(2,210,39,156,52)(3,53,157,40,211)(4,212,41,158,54)(5,55,159,42,213)(6,214,43,160,56)(7,57,145,44,215)(8,216,45,146,58)(9,59,147,46,217)(10,218,47,148,60)(11,61,149,48,219)(12,220,33,150,62)(13,63,151,34,221)(14,222,35,152,64)(15,49,153,36,223)(16,224,37,154,50)(17,251,114,186,226)(18,227,187,115,252)(19,253,116,188,228)(20,229,189,117,254)(21,255,118,190,230)(22,231,191,119,256)(23,241,120,192,232)(24,233,177,121,242)(25,243,122,178,234)(26,235,179,123,244)(27,245,124,180,236)(28,237,181,125,246)(29,247,126,182,238)(30,239,183,127,248)(31,249,128,184,240)(32,225,185,113,250)(65,195,134,291,99)(66,100,292,135,196)(67,197,136,293,101)(68,102,294,137,198)(69,199,138,295,103)(70,104,296,139,200)(71,201,140,297,105)(72,106,298,141,202)(73,203,142,299,107)(74,108,300,143,204)(75,205,144,301,109)(76,110,302,129,206)(77,207,130,303,111)(78,112,304,131,208)(79,193,132,289,97)(80,98,290,133,194)(81,261,170,286,311)(82,312,287,171,262)(83,263,172,288,313)(84,314,273,173,264)(85,265,174,274,315)(86,316,275,175,266)(87,267,176,276,317)(88,318,277,161,268)(89,269,162,278,319)(90,320,279,163,270)(91,271,164,280,305)(92,306,281,165,272)(93,257,166,282,307)(94,308,283,167,258)(95,259,168,284,309)(96,310,285,169,260), (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( (1,167,115,298)(2,168,116,299)(3,169,117,300)(4,170,118,301)(5,171,119,302)(6,172,120,303)(7,173,121,304)(8,174,122,289)(9,175,123,290)(10,176,124,291)(11,161,125,292)(12,162,126,293)(13,163,127,294)(14,164,128,295)(15,165,113,296)(16,166,114,297)(17,201,154,93)(18,202,155,94)(19,203,156,95)(20,204,157,96)(21,205,158,81)(22,206,159,82)(23,207,160,83)(24,208,145,84)(25,193,146,85)(26,194,147,86)(27,195,148,87)(28,196,149,88)(29,197,150,89)(30,198,151,90)(31,199,152,91)(32,200,153,92)(33,319,238,67)(34,320,239,68)(35,305,240,69)(36,306,225,70)(37,307,226,71)(38,308,227,72)(39,309,228,73)(40,310,229,74)(41,311,230,75)(42,312,231,76)(43,313,232,77)(44,314,233,78)(45,315,234,79)(46,316,235,80)(47,317,236,65)(48,318,237,66)(49,272,250,139)(50,257,251,140)(51,258,252,141)(52,259,253,142)(53,260,254,143)(54,261,255,144)(55,262,256,129)(56,263,241,130)(57,264,242,131)(58,265,243,132)(59,266,244,133)(60,267,245,134)(61,268,246,135)(62,269,247,136)(63,270,248,137)(64,271,249,138)(97,216,274,178)(98,217,275,179)(99,218,276,180)(100,219,277,181)(101,220,278,182)(102,221,279,183)(103,222,280,184)(104,223,281,185)(105,224,282,186)(106,209,283,187)(107,210,284,188)(108,211,285,189)(109,212,286,190)(110,213,287,191)(111,214,288,192)(112,215,273,177), (1,51,155,38,209)(2,210,39,156,52)(3,53,157,40,211)(4,212,41,158,54)(5,55,159,42,213)(6,214,43,160,56)(7,57,145,44,215)(8,216,45,146,58)(9,59,147,46,217)(10,218,47,148,60)(11,61,149,48,219)(12,220,33,150,62)(13,63,151,34,221)(14,222,35,152,64)(15,49,153,36,223)(16,224,37,154,50)(17,251,114,186,226)(18,227,187,115,252)(19,253,116,188,228)(20,229,189,117,254)(21,255,118,190,230)(22,231,191,119,256)(23,241,120,192,232)(24,233,177,121,242)(25,243,122,178,234)(26,235,179,123,244)(27,245,124,180,236)(28,237,181,125,246)(29,247,126,182,238)(30,239,183,127,248)(31,249,128,184,240)(32,225,185,113,250)(65,195,134,291,99)(66,100,292,135,196)(67,197,136,293,101)(68,102,294,137,198)(69,199,138,295,103)(70,104,296,139,200)(71,201,140,297,105)(72,106,298,141,202)(73,203,142,299,107)(74,108,300,143,204)(75,205,144,301,109)(76,110,302,129,206)(77,207,130,303,111)(78,112,304,131,208)(79,193,132,289,97)(80,98,290,133,194)(81,261,170,286,311)(82,312,287,171,262)(83,263,172,288,313)(84,314,273,173,264)(85,265,174,274,315)(86,316,275,175,266)(87,267,176,276,317)(88,318,277,161,268)(89,269,162,278,319)(90,320,279,163,270)(91,271,164,280,305)(92,306,281,165,272)(93,257,166,282,307)(94,308,283,167,258)(95,259,168,284,309)(96,310,285,169,260), (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([[(1,167,115,298),(2,168,116,299),(3,169,117,300),(4,170,118,301),(5,171,119,302),(6,172,120,303),(7,173,121,304),(8,174,122,289),(9,175,123,290),(10,176,124,291),(11,161,125,292),(12,162,126,293),(13,163,127,294),(14,164,128,295),(15,165,113,296),(16,166,114,297),(17,201,154,93),(18,202,155,94),(19,203,156,95),(20,204,157,96),(21,205,158,81),(22,206,159,82),(23,207,160,83),(24,208,145,84),(25,193,146,85),(26,194,147,86),(27,195,148,87),(28,196,149,88),(29,197,150,89),(30,198,151,90),(31,199,152,91),(32,200,153,92),(33,319,238,67),(34,320,239,68),(35,305,240,69),(36,306,225,70),(37,307,226,71),(38,308,227,72),(39,309,228,73),(40,310,229,74),(41,311,230,75),(42,312,231,76),(43,313,232,77),(44,314,233,78),(45,315,234,79),(46,316,235,80),(47,317,236,65),(48,318,237,66),(49,272,250,139),(50,257,251,140),(51,258,252,141),(52,259,253,142),(53,260,254,143),(54,261,255,144),(55,262,256,129),(56,263,241,130),(57,264,242,131),(58,265,243,132),(59,266,244,133),(60,267,245,134),(61,268,246,135),(62,269,247,136),(63,270,248,137),(64,271,249,138),(97,216,274,178),(98,217,275,179),(99,218,276,180),(100,219,277,181),(101,220,278,182),(102,221,279,183),(103,222,280,184),(104,223,281,185),(105,224,282,186),(106,209,283,187),(107,210,284,188),(108,211,285,189),(109,212,286,190),(110,213,287,191),(111,214,288,192),(112,215,273,177)], [(1,51,155,38,209),(2,210,39,156,52),(3,53,157,40,211),(4,212,41,158,54),(5,55,159,42,213),(6,214,43,160,56),(7,57,145,44,215),(8,216,45,146,58),(9,59,147,46,217),(10,218,47,148,60),(11,61,149,48,219),(12,220,33,150,62),(13,63,151,34,221),(14,222,35,152,64),(15,49,153,36,223),(16,224,37,154,50),(17,251,114,186,226),(18,227,187,115,252),(19,253,116,188,228),(20,229,189,117,254),(21,255,118,190,230),(22,231,191,119,256),(23,241,120,192,232),(24,233,177,121,242),(25,243,122,178,234),(26,235,179,123,244),(27,245,124,180,236),(28,237,181,125,246),(29,247,126,182,238),(30,239,183,127,248),(31,249,128,184,240),(32,225,185,113,250),(65,195,134,291,99),(66,100,292,135,196),(67,197,136,293,101),(68,102,294,137,198),(69,199,138,295,103),(70,104,296,139,200),(71,201,140,297,105),(72,106,298,141,202),(73,203,142,299,107),(74,108,300,143,204),(75,205,144,301,109),(76,110,302,129,206),(77,207,130,303,111),(78,112,304,131,208),(79,193,132,289,97),(80,98,290,133,194),(81,261,170,286,311),(82,312,287,171,262),(83,263,172,288,313),(84,314,273,173,264),(85,265,174,274,315),(86,316,275,175,266),(87,267,176,276,317),(88,318,277,161,268),(89,269,162,278,319),(90,320,279,163,270),(91,271,164,280,305),(92,306,281,165,272),(93,257,166,282,307),(94,308,283,167,258),(95,259,168,284,309),(96,310,285,169,260)], [(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)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 8A | ··· | 8P | 10A | ··· | 10F | 16A | ··· | 16AF | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 | D5 | Dic5 | Dic5 | D10 | C5⋊2C8 | C4×D5 | C5⋊2C8 | C5⋊2C16 |
| kernel | C4×C5⋊2C16 | C2×C5⋊2C16 | C4×C40 | C5⋊2C16 | C4×C20 | C2×C40 | C40 | C2×C20 | C20 | C4×C8 | C42 | C2×C8 | C2×C8 | C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 8 | 8 | 32 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 32 |
Matrix representation of C4×C5⋊2C16 ►in GL3(𝔽241) generated by
| 177 | 0 | 0 |
| 0 | 240 | 0 |
| 0 | 0 | 240 |
| 1 | 0 | 0 |
| 0 | 189 | 240 |
| 0 | 1 | 0 |
| 240 | 0 | 0 |
| 0 | 144 | 32 |
| 0 | 15 | 97 |
G:=sub<GL(3,GF(241))| [177,0,0,0,240,0,0,0,240],[1,0,0,0,189,1,0,240,0],[240,0,0,0,144,15,0,32,97] >;
C4×C5⋊2C16 in GAP, Magma, Sage, TeX
C_4\times C_5\rtimes_2C_{16} % in TeX
G:=Group("C4xC5:2C16"); // GroupNames label
G:=SmallGroup(320,18);
// by ID
G=gap.SmallGroup(320,18);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,100,102,12550]);
// 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^-1>;
// generators/relations
Export