direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C29, C3⋊C116, C87⋊5C4, C6.C58, C58.2S3, C174.3C2, C2.(S3×C29), SmallGroup(348,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C29 |
Generators and relations for Dic3×C29
G = < a,b,c | a29=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C29
►Regular action on 348 points
Generators in S348
(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)(320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348)
(1 347 291 199 241 147)(2 348 292 200 242 148)(3 320 293 201 243 149)(4 321 294 202 244 150)(5 322 295 203 245 151)(6 323 296 175 246 152)(7 324 297 176 247 153)(8 325 298 177 248 154)(9 326 299 178 249 155)(10 327 300 179 250 156)(11 328 301 180 251 157)(12 329 302 181 252 158)(13 330 303 182 253 159)(14 331 304 183 254 160)(15 332 305 184 255 161)(16 333 306 185 256 162)(17 334 307 186 257 163)(18 335 308 187 258 164)(19 336 309 188 259 165)(20 337 310 189 260 166)(21 338 311 190 261 167)(22 339 312 191 233 168)(23 340 313 192 234 169)(24 341 314 193 235 170)(25 342 315 194 236 171)(26 343 316 195 237 172)(27 344 317 196 238 173)(28 345 318 197 239 174)(29 346 319 198 240 146)(30 132 279 231 92 80)(31 133 280 232 93 81)(32 134 281 204 94 82)(33 135 282 205 95 83)(34 136 283 206 96 84)(35 137 284 207 97 85)(36 138 285 208 98 86)(37 139 286 209 99 87)(38 140 287 210 100 59)(39 141 288 211 101 60)(40 142 289 212 102 61)(41 143 290 213 103 62)(42 144 262 214 104 63)(43 145 263 215 105 64)(44 117 264 216 106 65)(45 118 265 217 107 66)(46 119 266 218 108 67)(47 120 267 219 109 68)(48 121 268 220 110 69)(49 122 269 221 111 70)(50 123 270 222 112 71)(51 124 271 223 113 72)(52 125 272 224 114 73)(53 126 273 225 115 74)(54 127 274 226 116 75)(55 128 275 227 88 76)(56 129 276 228 89 77)(57 130 277 229 90 78)(58 131 278 230 91 79)
(1 215 199 43)(2 216 200 44)(3 217 201 45)(4 218 202 46)(5 219 203 47)(6 220 175 48)(7 221 176 49)(8 222 177 50)(9 223 178 51)(10 224 179 52)(11 225 180 53)(12 226 181 54)(13 227 182 55)(14 228 183 56)(15 229 184 57)(16 230 185 58)(17 231 186 30)(18 232 187 31)(19 204 188 32)(20 205 189 33)(21 206 190 34)(22 207 191 35)(23 208 192 36)(24 209 193 37)(25 210 194 38)(26 211 195 39)(27 212 196 40)(28 213 197 41)(29 214 198 42)(59 342 287 236)(60 343 288 237)(61 344 289 238)(62 345 290 239)(63 346 262 240)(64 347 263 241)(65 348 264 242)(66 320 265 243)(67 321 266 244)(68 322 267 245)(69 323 268 246)(70 324 269 247)(71 325 270 248)(72 326 271 249)(73 327 272 250)(74 328 273 251)(75 329 274 252)(76 330 275 253)(77 331 276 254)(78 332 277 255)(79 333 278 256)(80 334 279 257)(81 335 280 258)(82 336 281 259)(83 337 282 260)(84 338 283 261)(85 339 284 233)(86 340 285 234)(87 341 286 235)(88 303 128 159)(89 304 129 160)(90 305 130 161)(91 306 131 162)(92 307 132 163)(93 308 133 164)(94 309 134 165)(95 310 135 166)(96 311 136 167)(97 312 137 168)(98 313 138 169)(99 314 139 170)(100 315 140 171)(101 316 141 172)(102 317 142 173)(103 318 143 174)(104 319 144 146)(105 291 145 147)(106 292 117 148)(107 293 118 149)(108 294 119 150)(109 295 120 151)(110 296 121 152)(111 297 122 153)(112 298 123 154)(113 299 124 155)(114 300 125 156)(115 301 126 157)(116 302 127 158)
G:=sub<Sym(348)| (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)(320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348), (1,347,291,199,241,147)(2,348,292,200,242,148)(3,320,293,201,243,149)(4,321,294,202,244,150)(5,322,295,203,245,151)(6,323,296,175,246,152)(7,324,297,176,247,153)(8,325,298,177,248,154)(9,326,299,178,249,155)(10,327,300,179,250,156)(11,328,301,180,251,157)(12,329,302,181,252,158)(13,330,303,182,253,159)(14,331,304,183,254,160)(15,332,305,184,255,161)(16,333,306,185,256,162)(17,334,307,186,257,163)(18,335,308,187,258,164)(19,336,309,188,259,165)(20,337,310,189,260,166)(21,338,311,190,261,167)(22,339,312,191,233,168)(23,340,313,192,234,169)(24,341,314,193,235,170)(25,342,315,194,236,171)(26,343,316,195,237,172)(27,344,317,196,238,173)(28,345,318,197,239,174)(29,346,319,198,240,146)(30,132,279,231,92,80)(31,133,280,232,93,81)(32,134,281,204,94,82)(33,135,282,205,95,83)(34,136,283,206,96,84)(35,137,284,207,97,85)(36,138,285,208,98,86)(37,139,286,209,99,87)(38,140,287,210,100,59)(39,141,288,211,101,60)(40,142,289,212,102,61)(41,143,290,213,103,62)(42,144,262,214,104,63)(43,145,263,215,105,64)(44,117,264,216,106,65)(45,118,265,217,107,66)(46,119,266,218,108,67)(47,120,267,219,109,68)(48,121,268,220,110,69)(49,122,269,221,111,70)(50,123,270,222,112,71)(51,124,271,223,113,72)(52,125,272,224,114,73)(53,126,273,225,115,74)(54,127,274,226,116,75)(55,128,275,227,88,76)(56,129,276,228,89,77)(57,130,277,229,90,78)(58,131,278,230,91,79), (1,215,199,43)(2,216,200,44)(3,217,201,45)(4,218,202,46)(5,219,203,47)(6,220,175,48)(7,221,176,49)(8,222,177,50)(9,223,178,51)(10,224,179,52)(11,225,180,53)(12,226,181,54)(13,227,182,55)(14,228,183,56)(15,229,184,57)(16,230,185,58)(17,231,186,30)(18,232,187,31)(19,204,188,32)(20,205,189,33)(21,206,190,34)(22,207,191,35)(23,208,192,36)(24,209,193,37)(25,210,194,38)(26,211,195,39)(27,212,196,40)(28,213,197,41)(29,214,198,42)(59,342,287,236)(60,343,288,237)(61,344,289,238)(62,345,290,239)(63,346,262,240)(64,347,263,241)(65,348,264,242)(66,320,265,243)(67,321,266,244)(68,322,267,245)(69,323,268,246)(70,324,269,247)(71,325,270,248)(72,326,271,249)(73,327,272,250)(74,328,273,251)(75,329,274,252)(76,330,275,253)(77,331,276,254)(78,332,277,255)(79,333,278,256)(80,334,279,257)(81,335,280,258)(82,336,281,259)(83,337,282,260)(84,338,283,261)(85,339,284,233)(86,340,285,234)(87,341,286,235)(88,303,128,159)(89,304,129,160)(90,305,130,161)(91,306,131,162)(92,307,132,163)(93,308,133,164)(94,309,134,165)(95,310,135,166)(96,311,136,167)(97,312,137,168)(98,313,138,169)(99,314,139,170)(100,315,140,171)(101,316,141,172)(102,317,142,173)(103,318,143,174)(104,319,144,146)(105,291,145,147)(106,292,117,148)(107,293,118,149)(108,294,119,150)(109,295,120,151)(110,296,121,152)(111,297,122,153)(112,298,123,154)(113,299,124,155)(114,300,125,156)(115,301,126,157)(116,302,127,158)>;
G:=Group( (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)(320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348), (1,347,291,199,241,147)(2,348,292,200,242,148)(3,320,293,201,243,149)(4,321,294,202,244,150)(5,322,295,203,245,151)(6,323,296,175,246,152)(7,324,297,176,247,153)(8,325,298,177,248,154)(9,326,299,178,249,155)(10,327,300,179,250,156)(11,328,301,180,251,157)(12,329,302,181,252,158)(13,330,303,182,253,159)(14,331,304,183,254,160)(15,332,305,184,255,161)(16,333,306,185,256,162)(17,334,307,186,257,163)(18,335,308,187,258,164)(19,336,309,188,259,165)(20,337,310,189,260,166)(21,338,311,190,261,167)(22,339,312,191,233,168)(23,340,313,192,234,169)(24,341,314,193,235,170)(25,342,315,194,236,171)(26,343,316,195,237,172)(27,344,317,196,238,173)(28,345,318,197,239,174)(29,346,319,198,240,146)(30,132,279,231,92,80)(31,133,280,232,93,81)(32,134,281,204,94,82)(33,135,282,205,95,83)(34,136,283,206,96,84)(35,137,284,207,97,85)(36,138,285,208,98,86)(37,139,286,209,99,87)(38,140,287,210,100,59)(39,141,288,211,101,60)(40,142,289,212,102,61)(41,143,290,213,103,62)(42,144,262,214,104,63)(43,145,263,215,105,64)(44,117,264,216,106,65)(45,118,265,217,107,66)(46,119,266,218,108,67)(47,120,267,219,109,68)(48,121,268,220,110,69)(49,122,269,221,111,70)(50,123,270,222,112,71)(51,124,271,223,113,72)(52,125,272,224,114,73)(53,126,273,225,115,74)(54,127,274,226,116,75)(55,128,275,227,88,76)(56,129,276,228,89,77)(57,130,277,229,90,78)(58,131,278,230,91,79), (1,215,199,43)(2,216,200,44)(3,217,201,45)(4,218,202,46)(5,219,203,47)(6,220,175,48)(7,221,176,49)(8,222,177,50)(9,223,178,51)(10,224,179,52)(11,225,180,53)(12,226,181,54)(13,227,182,55)(14,228,183,56)(15,229,184,57)(16,230,185,58)(17,231,186,30)(18,232,187,31)(19,204,188,32)(20,205,189,33)(21,206,190,34)(22,207,191,35)(23,208,192,36)(24,209,193,37)(25,210,194,38)(26,211,195,39)(27,212,196,40)(28,213,197,41)(29,214,198,42)(59,342,287,236)(60,343,288,237)(61,344,289,238)(62,345,290,239)(63,346,262,240)(64,347,263,241)(65,348,264,242)(66,320,265,243)(67,321,266,244)(68,322,267,245)(69,323,268,246)(70,324,269,247)(71,325,270,248)(72,326,271,249)(73,327,272,250)(74,328,273,251)(75,329,274,252)(76,330,275,253)(77,331,276,254)(78,332,277,255)(79,333,278,256)(80,334,279,257)(81,335,280,258)(82,336,281,259)(83,337,282,260)(84,338,283,261)(85,339,284,233)(86,340,285,234)(87,341,286,235)(88,303,128,159)(89,304,129,160)(90,305,130,161)(91,306,131,162)(92,307,132,163)(93,308,133,164)(94,309,134,165)(95,310,135,166)(96,311,136,167)(97,312,137,168)(98,313,138,169)(99,314,139,170)(100,315,140,171)(101,316,141,172)(102,317,142,173)(103,318,143,174)(104,319,144,146)(105,291,145,147)(106,292,117,148)(107,293,118,149)(108,294,119,150)(109,295,120,151)(110,296,121,152)(111,297,122,153)(112,298,123,154)(113,299,124,155)(114,300,125,156)(115,301,126,157)(116,302,127,158) );
G=PermutationGroup([[(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),(320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348)], [(1,347,291,199,241,147),(2,348,292,200,242,148),(3,320,293,201,243,149),(4,321,294,202,244,150),(5,322,295,203,245,151),(6,323,296,175,246,152),(7,324,297,176,247,153),(8,325,298,177,248,154),(9,326,299,178,249,155),(10,327,300,179,250,156),(11,328,301,180,251,157),(12,329,302,181,252,158),(13,330,303,182,253,159),(14,331,304,183,254,160),(15,332,305,184,255,161),(16,333,306,185,256,162),(17,334,307,186,257,163),(18,335,308,187,258,164),(19,336,309,188,259,165),(20,337,310,189,260,166),(21,338,311,190,261,167),(22,339,312,191,233,168),(23,340,313,192,234,169),(24,341,314,193,235,170),(25,342,315,194,236,171),(26,343,316,195,237,172),(27,344,317,196,238,173),(28,345,318,197,239,174),(29,346,319,198,240,146),(30,132,279,231,92,80),(31,133,280,232,93,81),(32,134,281,204,94,82),(33,135,282,205,95,83),(34,136,283,206,96,84),(35,137,284,207,97,85),(36,138,285,208,98,86),(37,139,286,209,99,87),(38,140,287,210,100,59),(39,141,288,211,101,60),(40,142,289,212,102,61),(41,143,290,213,103,62),(42,144,262,214,104,63),(43,145,263,215,105,64),(44,117,264,216,106,65),(45,118,265,217,107,66),(46,119,266,218,108,67),(47,120,267,219,109,68),(48,121,268,220,110,69),(49,122,269,221,111,70),(50,123,270,222,112,71),(51,124,271,223,113,72),(52,125,272,224,114,73),(53,126,273,225,115,74),(54,127,274,226,116,75),(55,128,275,227,88,76),(56,129,276,228,89,77),(57,130,277,229,90,78),(58,131,278,230,91,79)], [(1,215,199,43),(2,216,200,44),(3,217,201,45),(4,218,202,46),(5,219,203,47),(6,220,175,48),(7,221,176,49),(8,222,177,50),(9,223,178,51),(10,224,179,52),(11,225,180,53),(12,226,181,54),(13,227,182,55),(14,228,183,56),(15,229,184,57),(16,230,185,58),(17,231,186,30),(18,232,187,31),(19,204,188,32),(20,205,189,33),(21,206,190,34),(22,207,191,35),(23,208,192,36),(24,209,193,37),(25,210,194,38),(26,211,195,39),(27,212,196,40),(28,213,197,41),(29,214,198,42),(59,342,287,236),(60,343,288,237),(61,344,289,238),(62,345,290,239),(63,346,262,240),(64,347,263,241),(65,348,264,242),(66,320,265,243),(67,321,266,244),(68,322,267,245),(69,323,268,246),(70,324,269,247),(71,325,270,248),(72,326,271,249),(73,327,272,250),(74,328,273,251),(75,329,274,252),(76,330,275,253),(77,331,276,254),(78,332,277,255),(79,333,278,256),(80,334,279,257),(81,335,280,258),(82,336,281,259),(83,337,282,260),(84,338,283,261),(85,339,284,233),(86,340,285,234),(87,341,286,235),(88,303,128,159),(89,304,129,160),(90,305,130,161),(91,306,131,162),(92,307,132,163),(93,308,133,164),(94,309,134,165),(95,310,135,166),(96,311,136,167),(97,312,137,168),(98,313,138,169),(99,314,139,170),(100,315,140,171),(101,316,141,172),(102,317,142,173),(103,318,143,174),(104,319,144,146),(105,291,145,147),(106,292,117,148),(107,293,118,149),(108,294,119,150),(109,295,120,151),(110,296,121,152),(111,297,122,153),(112,298,123,154),(113,299,124,155),(114,300,125,156),(115,301,126,157),(116,302,127,158)]])
174 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 29A | ··· | 29AB | 58A | ··· | 58AB | 87A | ··· | 87AB | 116A | ··· | 116BD | 174A | ··· | 174AB |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 29 | ··· | 29 | 58 | ··· | 58 | 87 | ··· | 87 | 116 | ··· | 116 | 174 | ··· | 174 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
174 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C29 | C58 | C116 | S3 | Dic3 | S3×C29 | Dic3×C29 |
| kernel | Dic3×C29 | C174 | C87 | Dic3 | C6 | C3 | C58 | C29 | C2 | C1 |
| # reps | 1 | 1 | 2 | 28 | 28 | 56 | 1 | 1 | 28 | 28 |
Matrix representation of Dic3×C29 ►in GL3(𝔽349) generated by
| 1 | 0 | 0 |
| 0 | 269 | 0 |
| 0 | 0 | 269 |
| 348 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 348 | 348 |
| 136 | 0 | 0 |
| 0 | 34 | 115 |
| 0 | 81 | 315 |
G:=sub<GL(3,GF(349))| [1,0,0,0,269,0,0,0,269],[348,0,0,0,0,348,0,1,348],[136,0,0,0,34,81,0,115,315] >;
Dic3×C29 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{29} % in TeX
G:=Group("Dic3xC29"); // GroupNames label
G:=SmallGroup(348,1);
// by ID
G=gap.SmallGroup(348,1);
# by ID
G:=PCGroup([4,-2,-29,-2,-3,232,3715]);
// Polycyclic
G:=Group<a,b,c|a^29=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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