Copied to
clipboard

G = Dic3×C29order 348 = 22·3·29

Direct product of C29 and Dic3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic3×C29, C3⋊C116, C875C4, C6.C58, C58.2S3, C174.3C2, C2.(S3×C29), SmallGroup(348,1)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C29
C1C3C6C174 — Dic3×C29
C3 — Dic3×C29
C1C58

Generators and relations for Dic3×C29
 G = < a,b,c | a29=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C116

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)
(1 174 336 221 183 292)(2 146 337 222 184 293)(3 147 338 223 185 294)(4 148 339 224 186 295)(5 149 340 225 187 296)(6 150 341 226 188 297)(7 151 342 227 189 298)(8 152 343 228 190 299)(9 153 344 229 191 300)(10 154 345 230 192 301)(11 155 346 231 193 302)(12 156 347 232 194 303)(13 157 348 204 195 304)(14 158 320 205 196 305)(15 159 321 206 197 306)(16 160 322 207 198 307)(17 161 323 208 199 308)(18 162 324 209 200 309)(19 163 325 210 201 310)(20 164 326 211 202 311)(21 165 327 212 203 312)(22 166 328 213 175 313)(23 167 329 214 176 314)(24 168 330 215 177 315)(25 169 331 216 178 316)(26 170 332 217 179 317)(27 171 333 218 180 318)(28 172 334 219 181 319)(29 173 335 220 182 291)(30 290 135 86 104 251)(31 262 136 87 105 252)(32 263 137 59 106 253)(33 264 138 60 107 254)(34 265 139 61 108 255)(35 266 140 62 109 256)(36 267 141 63 110 257)(37 268 142 64 111 258)(38 269 143 65 112 259)(39 270 144 66 113 260)(40 271 145 67 114 261)(41 272 117 68 115 233)(42 273 118 69 116 234)(43 274 119 70 88 235)(44 275 120 71 89 236)(45 276 121 72 90 237)(46 277 122 73 91 238)(47 278 123 74 92 239)(48 279 124 75 93 240)(49 280 125 76 94 241)(50 281 126 77 95 242)(51 282 127 78 96 243)(52 283 128 79 97 244)(53 284 129 80 98 245)(54 285 130 81 99 246)(55 286 131 82 100 247)(56 287 132 83 101 248)(57 288 133 84 102 249)(58 289 134 85 103 250)
(1 86 221 30)(2 87 222 31)(3 59 223 32)(4 60 224 33)(5 61 225 34)(6 62 226 35)(7 63 227 36)(8 64 228 37)(9 65 229 38)(10 66 230 39)(11 67 231 40)(12 68 232 41)(13 69 204 42)(14 70 205 43)(15 71 206 44)(16 72 207 45)(17 73 208 46)(18 74 209 47)(19 75 210 48)(20 76 211 49)(21 77 212 50)(22 78 213 51)(23 79 214 52)(24 80 215 53)(25 81 216 54)(26 82 217 55)(27 83 218 56)(28 84 219 57)(29 85 220 58)(88 320 274 305)(89 321 275 306)(90 322 276 307)(91 323 277 308)(92 324 278 309)(93 325 279 310)(94 326 280 311)(95 327 281 312)(96 328 282 313)(97 329 283 314)(98 330 284 315)(99 331 285 316)(100 332 286 317)(101 333 287 318)(102 334 288 319)(103 335 289 291)(104 336 290 292)(105 337 262 293)(106 338 263 294)(107 339 264 295)(108 340 265 296)(109 341 266 297)(110 342 267 298)(111 343 268 299)(112 344 269 300)(113 345 270 301)(114 346 271 302)(115 347 272 303)(116 348 273 304)(117 194 233 156)(118 195 234 157)(119 196 235 158)(120 197 236 159)(121 198 237 160)(122 199 238 161)(123 200 239 162)(124 201 240 163)(125 202 241 164)(126 203 242 165)(127 175 243 166)(128 176 244 167)(129 177 245 168)(130 178 246 169)(131 179 247 170)(132 180 248 171)(133 181 249 172)(134 182 250 173)(135 183 251 174)(136 184 252 146)(137 185 253 147)(138 186 254 148)(139 187 255 149)(140 188 256 150)(141 189 257 151)(142 190 258 152)(143 191 259 153)(144 192 260 154)(145 193 261 155)

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), (1,174,336,221,183,292)(2,146,337,222,184,293)(3,147,338,223,185,294)(4,148,339,224,186,295)(5,149,340,225,187,296)(6,150,341,226,188,297)(7,151,342,227,189,298)(8,152,343,228,190,299)(9,153,344,229,191,300)(10,154,345,230,192,301)(11,155,346,231,193,302)(12,156,347,232,194,303)(13,157,348,204,195,304)(14,158,320,205,196,305)(15,159,321,206,197,306)(16,160,322,207,198,307)(17,161,323,208,199,308)(18,162,324,209,200,309)(19,163,325,210,201,310)(20,164,326,211,202,311)(21,165,327,212,203,312)(22,166,328,213,175,313)(23,167,329,214,176,314)(24,168,330,215,177,315)(25,169,331,216,178,316)(26,170,332,217,179,317)(27,171,333,218,180,318)(28,172,334,219,181,319)(29,173,335,220,182,291)(30,290,135,86,104,251)(31,262,136,87,105,252)(32,263,137,59,106,253)(33,264,138,60,107,254)(34,265,139,61,108,255)(35,266,140,62,109,256)(36,267,141,63,110,257)(37,268,142,64,111,258)(38,269,143,65,112,259)(39,270,144,66,113,260)(40,271,145,67,114,261)(41,272,117,68,115,233)(42,273,118,69,116,234)(43,274,119,70,88,235)(44,275,120,71,89,236)(45,276,121,72,90,237)(46,277,122,73,91,238)(47,278,123,74,92,239)(48,279,124,75,93,240)(49,280,125,76,94,241)(50,281,126,77,95,242)(51,282,127,78,96,243)(52,283,128,79,97,244)(53,284,129,80,98,245)(54,285,130,81,99,246)(55,286,131,82,100,247)(56,287,132,83,101,248)(57,288,133,84,102,249)(58,289,134,85,103,250), (1,86,221,30)(2,87,222,31)(3,59,223,32)(4,60,224,33)(5,61,225,34)(6,62,226,35)(7,63,227,36)(8,64,228,37)(9,65,229,38)(10,66,230,39)(11,67,231,40)(12,68,232,41)(13,69,204,42)(14,70,205,43)(15,71,206,44)(16,72,207,45)(17,73,208,46)(18,74,209,47)(19,75,210,48)(20,76,211,49)(21,77,212,50)(22,78,213,51)(23,79,214,52)(24,80,215,53)(25,81,216,54)(26,82,217,55)(27,83,218,56)(28,84,219,57)(29,85,220,58)(88,320,274,305)(89,321,275,306)(90,322,276,307)(91,323,277,308)(92,324,278,309)(93,325,279,310)(94,326,280,311)(95,327,281,312)(96,328,282,313)(97,329,283,314)(98,330,284,315)(99,331,285,316)(100,332,286,317)(101,333,287,318)(102,334,288,319)(103,335,289,291)(104,336,290,292)(105,337,262,293)(106,338,263,294)(107,339,264,295)(108,340,265,296)(109,341,266,297)(110,342,267,298)(111,343,268,299)(112,344,269,300)(113,345,270,301)(114,346,271,302)(115,347,272,303)(116,348,273,304)(117,194,233,156)(118,195,234,157)(119,196,235,158)(120,197,236,159)(121,198,237,160)(122,199,238,161)(123,200,239,162)(124,201,240,163)(125,202,241,164)(126,203,242,165)(127,175,243,166)(128,176,244,167)(129,177,245,168)(130,178,246,169)(131,179,247,170)(132,180,248,171)(133,181,249,172)(134,182,250,173)(135,183,251,174)(136,184,252,146)(137,185,253,147)(138,186,254,148)(139,187,255,149)(140,188,256,150)(141,189,257,151)(142,190,258,152)(143,191,259,153)(144,192,260,154)(145,193,261,155)>;

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), (1,174,336,221,183,292)(2,146,337,222,184,293)(3,147,338,223,185,294)(4,148,339,224,186,295)(5,149,340,225,187,296)(6,150,341,226,188,297)(7,151,342,227,189,298)(8,152,343,228,190,299)(9,153,344,229,191,300)(10,154,345,230,192,301)(11,155,346,231,193,302)(12,156,347,232,194,303)(13,157,348,204,195,304)(14,158,320,205,196,305)(15,159,321,206,197,306)(16,160,322,207,198,307)(17,161,323,208,199,308)(18,162,324,209,200,309)(19,163,325,210,201,310)(20,164,326,211,202,311)(21,165,327,212,203,312)(22,166,328,213,175,313)(23,167,329,214,176,314)(24,168,330,215,177,315)(25,169,331,216,178,316)(26,170,332,217,179,317)(27,171,333,218,180,318)(28,172,334,219,181,319)(29,173,335,220,182,291)(30,290,135,86,104,251)(31,262,136,87,105,252)(32,263,137,59,106,253)(33,264,138,60,107,254)(34,265,139,61,108,255)(35,266,140,62,109,256)(36,267,141,63,110,257)(37,268,142,64,111,258)(38,269,143,65,112,259)(39,270,144,66,113,260)(40,271,145,67,114,261)(41,272,117,68,115,233)(42,273,118,69,116,234)(43,274,119,70,88,235)(44,275,120,71,89,236)(45,276,121,72,90,237)(46,277,122,73,91,238)(47,278,123,74,92,239)(48,279,124,75,93,240)(49,280,125,76,94,241)(50,281,126,77,95,242)(51,282,127,78,96,243)(52,283,128,79,97,244)(53,284,129,80,98,245)(54,285,130,81,99,246)(55,286,131,82,100,247)(56,287,132,83,101,248)(57,288,133,84,102,249)(58,289,134,85,103,250), (1,86,221,30)(2,87,222,31)(3,59,223,32)(4,60,224,33)(5,61,225,34)(6,62,226,35)(7,63,227,36)(8,64,228,37)(9,65,229,38)(10,66,230,39)(11,67,231,40)(12,68,232,41)(13,69,204,42)(14,70,205,43)(15,71,206,44)(16,72,207,45)(17,73,208,46)(18,74,209,47)(19,75,210,48)(20,76,211,49)(21,77,212,50)(22,78,213,51)(23,79,214,52)(24,80,215,53)(25,81,216,54)(26,82,217,55)(27,83,218,56)(28,84,219,57)(29,85,220,58)(88,320,274,305)(89,321,275,306)(90,322,276,307)(91,323,277,308)(92,324,278,309)(93,325,279,310)(94,326,280,311)(95,327,281,312)(96,328,282,313)(97,329,283,314)(98,330,284,315)(99,331,285,316)(100,332,286,317)(101,333,287,318)(102,334,288,319)(103,335,289,291)(104,336,290,292)(105,337,262,293)(106,338,263,294)(107,339,264,295)(108,340,265,296)(109,341,266,297)(110,342,267,298)(111,343,268,299)(112,344,269,300)(113,345,270,301)(114,346,271,302)(115,347,272,303)(116,348,273,304)(117,194,233,156)(118,195,234,157)(119,196,235,158)(120,197,236,159)(121,198,237,160)(122,199,238,161)(123,200,239,162)(124,201,240,163)(125,202,241,164)(126,203,242,165)(127,175,243,166)(128,176,244,167)(129,177,245,168)(130,178,246,169)(131,179,247,170)(132,180,248,171)(133,181,249,172)(134,182,250,173)(135,183,251,174)(136,184,252,146)(137,185,253,147)(138,186,254,148)(139,187,255,149)(140,188,256,150)(141,189,257,151)(142,190,258,152)(143,191,259,153)(144,192,260,154)(145,193,261,155) );

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)], [(1,174,336,221,183,292),(2,146,337,222,184,293),(3,147,338,223,185,294),(4,148,339,224,186,295),(5,149,340,225,187,296),(6,150,341,226,188,297),(7,151,342,227,189,298),(8,152,343,228,190,299),(9,153,344,229,191,300),(10,154,345,230,192,301),(11,155,346,231,193,302),(12,156,347,232,194,303),(13,157,348,204,195,304),(14,158,320,205,196,305),(15,159,321,206,197,306),(16,160,322,207,198,307),(17,161,323,208,199,308),(18,162,324,209,200,309),(19,163,325,210,201,310),(20,164,326,211,202,311),(21,165,327,212,203,312),(22,166,328,213,175,313),(23,167,329,214,176,314),(24,168,330,215,177,315),(25,169,331,216,178,316),(26,170,332,217,179,317),(27,171,333,218,180,318),(28,172,334,219,181,319),(29,173,335,220,182,291),(30,290,135,86,104,251),(31,262,136,87,105,252),(32,263,137,59,106,253),(33,264,138,60,107,254),(34,265,139,61,108,255),(35,266,140,62,109,256),(36,267,141,63,110,257),(37,268,142,64,111,258),(38,269,143,65,112,259),(39,270,144,66,113,260),(40,271,145,67,114,261),(41,272,117,68,115,233),(42,273,118,69,116,234),(43,274,119,70,88,235),(44,275,120,71,89,236),(45,276,121,72,90,237),(46,277,122,73,91,238),(47,278,123,74,92,239),(48,279,124,75,93,240),(49,280,125,76,94,241),(50,281,126,77,95,242),(51,282,127,78,96,243),(52,283,128,79,97,244),(53,284,129,80,98,245),(54,285,130,81,99,246),(55,286,131,82,100,247),(56,287,132,83,101,248),(57,288,133,84,102,249),(58,289,134,85,103,250)], [(1,86,221,30),(2,87,222,31),(3,59,223,32),(4,60,224,33),(5,61,225,34),(6,62,226,35),(7,63,227,36),(8,64,228,37),(9,65,229,38),(10,66,230,39),(11,67,231,40),(12,68,232,41),(13,69,204,42),(14,70,205,43),(15,71,206,44),(16,72,207,45),(17,73,208,46),(18,74,209,47),(19,75,210,48),(20,76,211,49),(21,77,212,50),(22,78,213,51),(23,79,214,52),(24,80,215,53),(25,81,216,54),(26,82,217,55),(27,83,218,56),(28,84,219,57),(29,85,220,58),(88,320,274,305),(89,321,275,306),(90,322,276,307),(91,323,277,308),(92,324,278,309),(93,325,279,310),(94,326,280,311),(95,327,281,312),(96,328,282,313),(97,329,283,314),(98,330,284,315),(99,331,285,316),(100,332,286,317),(101,333,287,318),(102,334,288,319),(103,335,289,291),(104,336,290,292),(105,337,262,293),(106,338,263,294),(107,339,264,295),(108,340,265,296),(109,341,266,297),(110,342,267,298),(111,343,268,299),(112,344,269,300),(113,345,270,301),(114,346,271,302),(115,347,272,303),(116,348,273,304),(117,194,233,156),(118,195,234,157),(119,196,235,158),(120,197,236,159),(121,198,237,160),(122,199,238,161),(123,200,239,162),(124,201,240,163),(125,202,241,164),(126,203,242,165),(127,175,243,166),(128,176,244,167),(129,177,245,168),(130,178,246,169),(131,179,247,170),(132,180,248,171),(133,181,249,172),(134,182,250,173),(135,183,251,174),(136,184,252,146),(137,185,253,147),(138,186,254,148),(139,187,255,149),(140,188,256,150),(141,189,257,151),(142,190,258,152),(143,191,259,153),(144,192,260,154),(145,193,261,155)])

174 conjugacy classes

class 1  2  3 4A4B 6 29A···29AB58A···58AB87A···87AB116A···116BD174A···174AB
order12344629···2958···5887···87116···116174···174
size1123321···11···12···23···32···2

174 irreducible representations

dim1111112222
type+++-
imageC1C2C4C29C58C116S3Dic3S3×C29Dic3×C29
kernelDic3×C29C174C87Dic3C6C3C58C29C2C1
# reps112282856112828

Matrix representation of Dic3×C29 in GL3(𝔽349) generated by

100
02690
00269
,
34800
001
0348348
,
13600
034115
081315
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

Export

Subgroup lattice of Dic3×C29 in TeX

׿
×
𝔽