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G = C5×C163C4order 320 = 26·5

Direct product of C5 and C163C4

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×C163C4, C8015C4, C163C20, C10.7Q32, C40.18Q8, C10.14D16, C20.18Q16, C8.2(C5×Q8), (C2×C80).9C2, C2.2(C5×D16), C2.2(C5×Q32), C4.1(C5×Q16), (C2×C16).3C10, C8.13(C2×C20), (C2×C10).51D8, C20.87(C4⋊C4), C2.D8.2C10, C40.122(C2×C4), (C2×C20).408D4, C22.10(C5×D8), C10.20(C2.D8), (C2×C40).418C22, C4.7(C5×C4⋊C4), C2.3(C5×C2.D8), (C2×C4).62(C5×D4), (C5×C2.D8).9C2, (C2×C8).73(C2×C10), SmallGroup(320,171)

Series: Derived Chief Lower central Upper central

C1C8 — C5×C163C4
C1C2C4C2×C4C2×C8C2×C40C5×C2.D8 — C5×C163C4
C1C2C4C8 — C5×C163C4
C1C2×C10C2×C20C2×C40 — C5×C163C4

Generators and relations for C5×C163C4
 G = < a,b,c | a5=b16=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

8C4
8C4
4C2×C4
4C2×C4
8C20
8C20
2C4⋊C4
2C4⋊C4
4C2×C20
4C2×C20
2C5×C4⋊C4
2C5×C4⋊C4

Smallest permutation representation of C5×C163C4
Regular action on 320 points
Generators in S320
(1 251 146 278 197)(2 252 147 279 198)(3 253 148 280 199)(4 254 149 281 200)(5 255 150 282 201)(6 256 151 283 202)(7 241 152 284 203)(8 242 153 285 204)(9 243 154 286 205)(10 244 155 287 206)(11 245 156 288 207)(12 246 157 273 208)(13 247 158 274 193)(14 248 159 275 194)(15 249 160 276 195)(16 250 145 277 196)(17 129 57 231 36)(18 130 58 232 37)(19 131 59 233 38)(20 132 60 234 39)(21 133 61 235 40)(22 134 62 236 41)(23 135 63 237 42)(24 136 64 238 43)(25 137 49 239 44)(26 138 50 240 45)(27 139 51 225 46)(28 140 52 226 47)(29 141 53 227 48)(30 142 54 228 33)(31 143 55 229 34)(32 144 56 230 35)(65 319 177 269 87)(66 320 178 270 88)(67 305 179 271 89)(68 306 180 272 90)(69 307 181 257 91)(70 308 182 258 92)(71 309 183 259 93)(72 310 184 260 94)(73 311 185 261 95)(74 312 186 262 96)(75 313 187 263 81)(76 314 188 264 82)(77 315 189 265 83)(78 316 190 266 84)(79 317 191 267 85)(80 318 192 268 86)(97 127 297 210 172)(98 128 298 211 173)(99 113 299 212 174)(100 114 300 213 175)(101 115 301 214 176)(102 116 302 215 161)(103 117 303 216 162)(104 118 304 217 163)(105 119 289 218 164)(106 120 290 219 165)(107 121 291 220 166)(108 122 292 221 167)(109 123 293 222 168)(110 124 294 223 169)(111 125 295 224 170)(112 126 296 209 171)
(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)
(1 262 44 106)(2 261 45 105)(3 260 46 104)(4 259 47 103)(5 258 48 102)(6 257 33 101)(7 272 34 100)(8 271 35 99)(9 270 36 98)(10 269 37 97)(11 268 38 112)(12 267 39 111)(13 266 40 110)(14 265 41 109)(15 264 42 108)(16 263 43 107)(17 128 243 88)(18 127 244 87)(19 126 245 86)(20 125 246 85)(21 124 247 84)(22 123 248 83)(23 122 249 82)(24 121 250 81)(25 120 251 96)(26 119 252 95)(27 118 253 94)(28 117 254 93)(29 116 255 92)(30 115 256 91)(31 114 241 90)(32 113 242 89)(49 219 278 312)(50 218 279 311)(51 217 280 310)(52 216 281 309)(53 215 282 308)(54 214 283 307)(55 213 284 306)(56 212 285 305)(57 211 286 320)(58 210 287 319)(59 209 288 318)(60 224 273 317)(61 223 274 316)(62 222 275 315)(63 221 276 314)(64 220 277 313)(65 130 297 155)(66 129 298 154)(67 144 299 153)(68 143 300 152)(69 142 301 151)(70 141 302 150)(71 140 303 149)(72 139 304 148)(73 138 289 147)(74 137 290 146)(75 136 291 145)(76 135 292 160)(77 134 293 159)(78 133 294 158)(79 132 295 157)(80 131 296 156)(161 201 182 227)(162 200 183 226)(163 199 184 225)(164 198 185 240)(165 197 186 239)(166 196 187 238)(167 195 188 237)(168 194 189 236)(169 193 190 235)(170 208 191 234)(171 207 192 233)(172 206 177 232)(173 205 178 231)(174 204 179 230)(175 203 180 229)(176 202 181 228)

G:=sub<Sym(320)| (1,251,146,278,197)(2,252,147,279,198)(3,253,148,280,199)(4,254,149,281,200)(5,255,150,282,201)(6,256,151,283,202)(7,241,152,284,203)(8,242,153,285,204)(9,243,154,286,205)(10,244,155,287,206)(11,245,156,288,207)(12,246,157,273,208)(13,247,158,274,193)(14,248,159,275,194)(15,249,160,276,195)(16,250,145,277,196)(17,129,57,231,36)(18,130,58,232,37)(19,131,59,233,38)(20,132,60,234,39)(21,133,61,235,40)(22,134,62,236,41)(23,135,63,237,42)(24,136,64,238,43)(25,137,49,239,44)(26,138,50,240,45)(27,139,51,225,46)(28,140,52,226,47)(29,141,53,227,48)(30,142,54,228,33)(31,143,55,229,34)(32,144,56,230,35)(65,319,177,269,87)(66,320,178,270,88)(67,305,179,271,89)(68,306,180,272,90)(69,307,181,257,91)(70,308,182,258,92)(71,309,183,259,93)(72,310,184,260,94)(73,311,185,261,95)(74,312,186,262,96)(75,313,187,263,81)(76,314,188,264,82)(77,315,189,265,83)(78,316,190,266,84)(79,317,191,267,85)(80,318,192,268,86)(97,127,297,210,172)(98,128,298,211,173)(99,113,299,212,174)(100,114,300,213,175)(101,115,301,214,176)(102,116,302,215,161)(103,117,303,216,162)(104,118,304,217,163)(105,119,289,218,164)(106,120,290,219,165)(107,121,291,220,166)(108,122,292,221,167)(109,123,293,222,168)(110,124,294,223,169)(111,125,295,224,170)(112,126,296,209,171), (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), (1,262,44,106)(2,261,45,105)(3,260,46,104)(4,259,47,103)(5,258,48,102)(6,257,33,101)(7,272,34,100)(8,271,35,99)(9,270,36,98)(10,269,37,97)(11,268,38,112)(12,267,39,111)(13,266,40,110)(14,265,41,109)(15,264,42,108)(16,263,43,107)(17,128,243,88)(18,127,244,87)(19,126,245,86)(20,125,246,85)(21,124,247,84)(22,123,248,83)(23,122,249,82)(24,121,250,81)(25,120,251,96)(26,119,252,95)(27,118,253,94)(28,117,254,93)(29,116,255,92)(30,115,256,91)(31,114,241,90)(32,113,242,89)(49,219,278,312)(50,218,279,311)(51,217,280,310)(52,216,281,309)(53,215,282,308)(54,214,283,307)(55,213,284,306)(56,212,285,305)(57,211,286,320)(58,210,287,319)(59,209,288,318)(60,224,273,317)(61,223,274,316)(62,222,275,315)(63,221,276,314)(64,220,277,313)(65,130,297,155)(66,129,298,154)(67,144,299,153)(68,143,300,152)(69,142,301,151)(70,141,302,150)(71,140,303,149)(72,139,304,148)(73,138,289,147)(74,137,290,146)(75,136,291,145)(76,135,292,160)(77,134,293,159)(78,133,294,158)(79,132,295,157)(80,131,296,156)(161,201,182,227)(162,200,183,226)(163,199,184,225)(164,198,185,240)(165,197,186,239)(166,196,187,238)(167,195,188,237)(168,194,189,236)(169,193,190,235)(170,208,191,234)(171,207,192,233)(172,206,177,232)(173,205,178,231)(174,204,179,230)(175,203,180,229)(176,202,181,228)>;

G:=Group( (1,251,146,278,197)(2,252,147,279,198)(3,253,148,280,199)(4,254,149,281,200)(5,255,150,282,201)(6,256,151,283,202)(7,241,152,284,203)(8,242,153,285,204)(9,243,154,286,205)(10,244,155,287,206)(11,245,156,288,207)(12,246,157,273,208)(13,247,158,274,193)(14,248,159,275,194)(15,249,160,276,195)(16,250,145,277,196)(17,129,57,231,36)(18,130,58,232,37)(19,131,59,233,38)(20,132,60,234,39)(21,133,61,235,40)(22,134,62,236,41)(23,135,63,237,42)(24,136,64,238,43)(25,137,49,239,44)(26,138,50,240,45)(27,139,51,225,46)(28,140,52,226,47)(29,141,53,227,48)(30,142,54,228,33)(31,143,55,229,34)(32,144,56,230,35)(65,319,177,269,87)(66,320,178,270,88)(67,305,179,271,89)(68,306,180,272,90)(69,307,181,257,91)(70,308,182,258,92)(71,309,183,259,93)(72,310,184,260,94)(73,311,185,261,95)(74,312,186,262,96)(75,313,187,263,81)(76,314,188,264,82)(77,315,189,265,83)(78,316,190,266,84)(79,317,191,267,85)(80,318,192,268,86)(97,127,297,210,172)(98,128,298,211,173)(99,113,299,212,174)(100,114,300,213,175)(101,115,301,214,176)(102,116,302,215,161)(103,117,303,216,162)(104,118,304,217,163)(105,119,289,218,164)(106,120,290,219,165)(107,121,291,220,166)(108,122,292,221,167)(109,123,293,222,168)(110,124,294,223,169)(111,125,295,224,170)(112,126,296,209,171), (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), (1,262,44,106)(2,261,45,105)(3,260,46,104)(4,259,47,103)(5,258,48,102)(6,257,33,101)(7,272,34,100)(8,271,35,99)(9,270,36,98)(10,269,37,97)(11,268,38,112)(12,267,39,111)(13,266,40,110)(14,265,41,109)(15,264,42,108)(16,263,43,107)(17,128,243,88)(18,127,244,87)(19,126,245,86)(20,125,246,85)(21,124,247,84)(22,123,248,83)(23,122,249,82)(24,121,250,81)(25,120,251,96)(26,119,252,95)(27,118,253,94)(28,117,254,93)(29,116,255,92)(30,115,256,91)(31,114,241,90)(32,113,242,89)(49,219,278,312)(50,218,279,311)(51,217,280,310)(52,216,281,309)(53,215,282,308)(54,214,283,307)(55,213,284,306)(56,212,285,305)(57,211,286,320)(58,210,287,319)(59,209,288,318)(60,224,273,317)(61,223,274,316)(62,222,275,315)(63,221,276,314)(64,220,277,313)(65,130,297,155)(66,129,298,154)(67,144,299,153)(68,143,300,152)(69,142,301,151)(70,141,302,150)(71,140,303,149)(72,139,304,148)(73,138,289,147)(74,137,290,146)(75,136,291,145)(76,135,292,160)(77,134,293,159)(78,133,294,158)(79,132,295,157)(80,131,296,156)(161,201,182,227)(162,200,183,226)(163,199,184,225)(164,198,185,240)(165,197,186,239)(166,196,187,238)(167,195,188,237)(168,194,189,236)(169,193,190,235)(170,208,191,234)(171,207,192,233)(172,206,177,232)(173,205,178,231)(174,204,179,230)(175,203,180,229)(176,202,181,228) );

G=PermutationGroup([[(1,251,146,278,197),(2,252,147,279,198),(3,253,148,280,199),(4,254,149,281,200),(5,255,150,282,201),(6,256,151,283,202),(7,241,152,284,203),(8,242,153,285,204),(9,243,154,286,205),(10,244,155,287,206),(11,245,156,288,207),(12,246,157,273,208),(13,247,158,274,193),(14,248,159,275,194),(15,249,160,276,195),(16,250,145,277,196),(17,129,57,231,36),(18,130,58,232,37),(19,131,59,233,38),(20,132,60,234,39),(21,133,61,235,40),(22,134,62,236,41),(23,135,63,237,42),(24,136,64,238,43),(25,137,49,239,44),(26,138,50,240,45),(27,139,51,225,46),(28,140,52,226,47),(29,141,53,227,48),(30,142,54,228,33),(31,143,55,229,34),(32,144,56,230,35),(65,319,177,269,87),(66,320,178,270,88),(67,305,179,271,89),(68,306,180,272,90),(69,307,181,257,91),(70,308,182,258,92),(71,309,183,259,93),(72,310,184,260,94),(73,311,185,261,95),(74,312,186,262,96),(75,313,187,263,81),(76,314,188,264,82),(77,315,189,265,83),(78,316,190,266,84),(79,317,191,267,85),(80,318,192,268,86),(97,127,297,210,172),(98,128,298,211,173),(99,113,299,212,174),(100,114,300,213,175),(101,115,301,214,176),(102,116,302,215,161),(103,117,303,216,162),(104,118,304,217,163),(105,119,289,218,164),(106,120,290,219,165),(107,121,291,220,166),(108,122,292,221,167),(109,123,293,222,168),(110,124,294,223,169),(111,125,295,224,170),(112,126,296,209,171)], [(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)], [(1,262,44,106),(2,261,45,105),(3,260,46,104),(4,259,47,103),(5,258,48,102),(6,257,33,101),(7,272,34,100),(8,271,35,99),(9,270,36,98),(10,269,37,97),(11,268,38,112),(12,267,39,111),(13,266,40,110),(14,265,41,109),(15,264,42,108),(16,263,43,107),(17,128,243,88),(18,127,244,87),(19,126,245,86),(20,125,246,85),(21,124,247,84),(22,123,248,83),(23,122,249,82),(24,121,250,81),(25,120,251,96),(26,119,252,95),(27,118,253,94),(28,117,254,93),(29,116,255,92),(30,115,256,91),(31,114,241,90),(32,113,242,89),(49,219,278,312),(50,218,279,311),(51,217,280,310),(52,216,281,309),(53,215,282,308),(54,214,283,307),(55,213,284,306),(56,212,285,305),(57,211,286,320),(58,210,287,319),(59,209,288,318),(60,224,273,317),(61,223,274,316),(62,222,275,315),(63,221,276,314),(64,220,277,313),(65,130,297,155),(66,129,298,154),(67,144,299,153),(68,143,300,152),(69,142,301,151),(70,141,302,150),(71,140,303,149),(72,139,304,148),(73,138,289,147),(74,137,290,146),(75,136,291,145),(76,135,292,160),(77,134,293,159),(78,133,294,158),(79,132,295,157),(80,131,296,156),(161,201,182,227),(162,200,183,226),(163,199,184,225),(164,198,185,240),(165,197,186,239),(166,196,187,238),(167,195,188,237),(168,194,189,236),(169,193,190,235),(170,208,191,234),(171,207,192,233),(172,206,177,232),(173,205,178,231),(174,204,179,230),(175,203,180,229),(176,202,181,228)]])

110 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B5C5D8A8B8C8D10A···10L16A···16H20A···20H20I···20X40A···40P80A···80AF
order12224444445555888810···1016···1620···2020···2040···4080···80
size1111228888111122221···12···22···28···82···22···2

110 irreducible representations

dim11111111222222222222
type+++-+-++-
imageC1C2C2C4C5C10C10C20Q8D4Q16D8D16Q32C5×Q8C5×D4C5×Q16C5×D8C5×D16C5×Q32
kernelC5×C163C4C5×C2.D8C2×C80C80C163C4C2.D8C2×C16C16C40C2×C20C20C2×C10C10C10C8C2×C4C4C22C2C2
# reps12144841611224444881616

Matrix representation of C5×C163C4 in GL3(𝔽241) generated by

100
0910
0091
,
24000
015627
0214156
,
6400
010650
050135
G:=sub<GL(3,GF(241))| [1,0,0,0,91,0,0,0,91],[240,0,0,0,156,214,0,27,156],[64,0,0,0,106,50,0,50,135] >;

C5×C163C4 in GAP, Magma, Sage, TeX

C_5\times C_{16}\rtimes_3C_4
% in TeX

G:=Group("C5xC16:3C4");
// GroupNames label

G:=SmallGroup(320,171);
// by ID

G=gap.SmallGroup(320,171);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,708,2803,360,10085,124]);
// 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^-1>;
// generators/relations

Export

Subgroup lattice of C5×C163C4 in TeX

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