direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C37, C3⋊C148, C6.C74, C111⋊5C4, C74.2S3, C222.3C2, C2.(S3×C37), SmallGroup(444,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C37 |
Generators and relations for Dic3×C37
G = < a,b,c | a37=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C37
►Regular action on 444 points
Generators in S444
(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)(334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370)(371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407)(408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444)
(1 49 412 253 99 122)(2 50 413 254 100 123)(3 51 414 255 101 124)(4 52 415 256 102 125)(5 53 416 257 103 126)(6 54 417 258 104 127)(7 55 418 259 105 128)(8 56 419 223 106 129)(9 57 420 224 107 130)(10 58 421 225 108 131)(11 59 422 226 109 132)(12 60 423 227 110 133)(13 61 424 228 111 134)(14 62 425 229 75 135)(15 63 426 230 76 136)(16 64 427 231 77 137)(17 65 428 232 78 138)(18 66 429 233 79 139)(19 67 430 234 80 140)(20 68 431 235 81 141)(21 69 432 236 82 142)(22 70 433 237 83 143)(23 71 434 238 84 144)(24 72 435 239 85 145)(25 73 436 240 86 146)(26 74 437 241 87 147)(27 38 438 242 88 148)(28 39 439 243 89 112)(29 40 440 244 90 113)(30 41 441 245 91 114)(31 42 442 246 92 115)(32 43 443 247 93 116)(33 44 444 248 94 117)(34 45 408 249 95 118)(35 46 409 250 96 119)(36 47 410 251 97 120)(37 48 411 252 98 121)(149 284 334 219 401 323)(150 285 335 220 402 324)(151 286 336 221 403 325)(152 287 337 222 404 326)(153 288 338 186 405 327)(154 289 339 187 406 328)(155 290 340 188 407 329)(156 291 341 189 371 330)(157 292 342 190 372 331)(158 293 343 191 373 332)(159 294 344 192 374 333)(160 295 345 193 375 297)(161 296 346 194 376 298)(162 260 347 195 377 299)(163 261 348 196 378 300)(164 262 349 197 379 301)(165 263 350 198 380 302)(166 264 351 199 381 303)(167 265 352 200 382 304)(168 266 353 201 383 305)(169 267 354 202 384 306)(170 268 355 203 385 307)(171 269 356 204 386 308)(172 270 357 205 387 309)(173 271 358 206 388 310)(174 272 359 207 389 311)(175 273 360 208 390 312)(176 274 361 209 391 313)(177 275 362 210 392 314)(178 276 363 211 393 315)(179 277 364 212 394 316)(180 278 365 213 395 317)(181 279 366 214 396 318)(182 280 367 215 397 319)(183 281 368 216 398 320)(184 282 369 217 399 321)(185 283 370 218 400 322)
(1 293 253 373)(2 294 254 374)(3 295 255 375)(4 296 256 376)(5 260 257 377)(6 261 258 378)(7 262 259 379)(8 263 223 380)(9 264 224 381)(10 265 225 382)(11 266 226 383)(12 267 227 384)(13 268 228 385)(14 269 229 386)(15 270 230 387)(16 271 231 388)(17 272 232 389)(18 273 233 390)(19 274 234 391)(20 275 235 392)(21 276 236 393)(22 277 237 394)(23 278 238 395)(24 279 239 396)(25 280 240 397)(26 281 241 398)(27 282 242 399)(28 283 243 400)(29 284 244 401)(30 285 245 402)(31 286 246 403)(32 287 247 404)(33 288 248 405)(34 289 249 406)(35 290 250 407)(36 291 251 371)(37 292 252 372)(38 184 88 217)(39 185 89 218)(40 149 90 219)(41 150 91 220)(42 151 92 221)(43 152 93 222)(44 153 94 186)(45 154 95 187)(46 155 96 188)(47 156 97 189)(48 157 98 190)(49 158 99 191)(50 159 100 192)(51 160 101 193)(52 161 102 194)(53 162 103 195)(54 163 104 196)(55 164 105 197)(56 165 106 198)(57 166 107 199)(58 167 108 200)(59 168 109 201)(60 169 110 202)(61 170 111 203)(62 171 75 204)(63 172 76 205)(64 173 77 206)(65 174 78 207)(66 175 79 208)(67 176 80 209)(68 177 81 210)(69 178 82 211)(70 179 83 212)(71 180 84 213)(72 181 85 214)(73 182 86 215)(74 183 87 216)(112 370 439 322)(113 334 440 323)(114 335 441 324)(115 336 442 325)(116 337 443 326)(117 338 444 327)(118 339 408 328)(119 340 409 329)(120 341 410 330)(121 342 411 331)(122 343 412 332)(123 344 413 333)(124 345 414 297)(125 346 415 298)(126 347 416 299)(127 348 417 300)(128 349 418 301)(129 350 419 302)(130 351 420 303)(131 352 421 304)(132 353 422 305)(133 354 423 306)(134 355 424 307)(135 356 425 308)(136 357 426 309)(137 358 427 310)(138 359 428 311)(139 360 429 312)(140 361 430 313)(141 362 431 314)(142 363 432 315)(143 364 433 316)(144 365 434 317)(145 366 435 318)(146 367 436 319)(147 368 437 320)(148 369 438 321)
G:=sub<Sym(444)| (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)(334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370)(371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407)(408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444), (1,49,412,253,99,122)(2,50,413,254,100,123)(3,51,414,255,101,124)(4,52,415,256,102,125)(5,53,416,257,103,126)(6,54,417,258,104,127)(7,55,418,259,105,128)(8,56,419,223,106,129)(9,57,420,224,107,130)(10,58,421,225,108,131)(11,59,422,226,109,132)(12,60,423,227,110,133)(13,61,424,228,111,134)(14,62,425,229,75,135)(15,63,426,230,76,136)(16,64,427,231,77,137)(17,65,428,232,78,138)(18,66,429,233,79,139)(19,67,430,234,80,140)(20,68,431,235,81,141)(21,69,432,236,82,142)(22,70,433,237,83,143)(23,71,434,238,84,144)(24,72,435,239,85,145)(25,73,436,240,86,146)(26,74,437,241,87,147)(27,38,438,242,88,148)(28,39,439,243,89,112)(29,40,440,244,90,113)(30,41,441,245,91,114)(31,42,442,246,92,115)(32,43,443,247,93,116)(33,44,444,248,94,117)(34,45,408,249,95,118)(35,46,409,250,96,119)(36,47,410,251,97,120)(37,48,411,252,98,121)(149,284,334,219,401,323)(150,285,335,220,402,324)(151,286,336,221,403,325)(152,287,337,222,404,326)(153,288,338,186,405,327)(154,289,339,187,406,328)(155,290,340,188,407,329)(156,291,341,189,371,330)(157,292,342,190,372,331)(158,293,343,191,373,332)(159,294,344,192,374,333)(160,295,345,193,375,297)(161,296,346,194,376,298)(162,260,347,195,377,299)(163,261,348,196,378,300)(164,262,349,197,379,301)(165,263,350,198,380,302)(166,264,351,199,381,303)(167,265,352,200,382,304)(168,266,353,201,383,305)(169,267,354,202,384,306)(170,268,355,203,385,307)(171,269,356,204,386,308)(172,270,357,205,387,309)(173,271,358,206,388,310)(174,272,359,207,389,311)(175,273,360,208,390,312)(176,274,361,209,391,313)(177,275,362,210,392,314)(178,276,363,211,393,315)(179,277,364,212,394,316)(180,278,365,213,395,317)(181,279,366,214,396,318)(182,280,367,215,397,319)(183,281,368,216,398,320)(184,282,369,217,399,321)(185,283,370,218,400,322), (1,293,253,373)(2,294,254,374)(3,295,255,375)(4,296,256,376)(5,260,257,377)(6,261,258,378)(7,262,259,379)(8,263,223,380)(9,264,224,381)(10,265,225,382)(11,266,226,383)(12,267,227,384)(13,268,228,385)(14,269,229,386)(15,270,230,387)(16,271,231,388)(17,272,232,389)(18,273,233,390)(19,274,234,391)(20,275,235,392)(21,276,236,393)(22,277,237,394)(23,278,238,395)(24,279,239,396)(25,280,240,397)(26,281,241,398)(27,282,242,399)(28,283,243,400)(29,284,244,401)(30,285,245,402)(31,286,246,403)(32,287,247,404)(33,288,248,405)(34,289,249,406)(35,290,250,407)(36,291,251,371)(37,292,252,372)(38,184,88,217)(39,185,89,218)(40,149,90,219)(41,150,91,220)(42,151,92,221)(43,152,93,222)(44,153,94,186)(45,154,95,187)(46,155,96,188)(47,156,97,189)(48,157,98,190)(49,158,99,191)(50,159,100,192)(51,160,101,193)(52,161,102,194)(53,162,103,195)(54,163,104,196)(55,164,105,197)(56,165,106,198)(57,166,107,199)(58,167,108,200)(59,168,109,201)(60,169,110,202)(61,170,111,203)(62,171,75,204)(63,172,76,205)(64,173,77,206)(65,174,78,207)(66,175,79,208)(67,176,80,209)(68,177,81,210)(69,178,82,211)(70,179,83,212)(71,180,84,213)(72,181,85,214)(73,182,86,215)(74,183,87,216)(112,370,439,322)(113,334,440,323)(114,335,441,324)(115,336,442,325)(116,337,443,326)(117,338,444,327)(118,339,408,328)(119,340,409,329)(120,341,410,330)(121,342,411,331)(122,343,412,332)(123,344,413,333)(124,345,414,297)(125,346,415,298)(126,347,416,299)(127,348,417,300)(128,349,418,301)(129,350,419,302)(130,351,420,303)(131,352,421,304)(132,353,422,305)(133,354,423,306)(134,355,424,307)(135,356,425,308)(136,357,426,309)(137,358,427,310)(138,359,428,311)(139,360,429,312)(140,361,430,313)(141,362,431,314)(142,363,432,315)(143,364,433,316)(144,365,434,317)(145,366,435,318)(146,367,436,319)(147,368,437,320)(148,369,438,321)>;
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)(334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370)(371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407)(408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444), (1,49,412,253,99,122)(2,50,413,254,100,123)(3,51,414,255,101,124)(4,52,415,256,102,125)(5,53,416,257,103,126)(6,54,417,258,104,127)(7,55,418,259,105,128)(8,56,419,223,106,129)(9,57,420,224,107,130)(10,58,421,225,108,131)(11,59,422,226,109,132)(12,60,423,227,110,133)(13,61,424,228,111,134)(14,62,425,229,75,135)(15,63,426,230,76,136)(16,64,427,231,77,137)(17,65,428,232,78,138)(18,66,429,233,79,139)(19,67,430,234,80,140)(20,68,431,235,81,141)(21,69,432,236,82,142)(22,70,433,237,83,143)(23,71,434,238,84,144)(24,72,435,239,85,145)(25,73,436,240,86,146)(26,74,437,241,87,147)(27,38,438,242,88,148)(28,39,439,243,89,112)(29,40,440,244,90,113)(30,41,441,245,91,114)(31,42,442,246,92,115)(32,43,443,247,93,116)(33,44,444,248,94,117)(34,45,408,249,95,118)(35,46,409,250,96,119)(36,47,410,251,97,120)(37,48,411,252,98,121)(149,284,334,219,401,323)(150,285,335,220,402,324)(151,286,336,221,403,325)(152,287,337,222,404,326)(153,288,338,186,405,327)(154,289,339,187,406,328)(155,290,340,188,407,329)(156,291,341,189,371,330)(157,292,342,190,372,331)(158,293,343,191,373,332)(159,294,344,192,374,333)(160,295,345,193,375,297)(161,296,346,194,376,298)(162,260,347,195,377,299)(163,261,348,196,378,300)(164,262,349,197,379,301)(165,263,350,198,380,302)(166,264,351,199,381,303)(167,265,352,200,382,304)(168,266,353,201,383,305)(169,267,354,202,384,306)(170,268,355,203,385,307)(171,269,356,204,386,308)(172,270,357,205,387,309)(173,271,358,206,388,310)(174,272,359,207,389,311)(175,273,360,208,390,312)(176,274,361,209,391,313)(177,275,362,210,392,314)(178,276,363,211,393,315)(179,277,364,212,394,316)(180,278,365,213,395,317)(181,279,366,214,396,318)(182,280,367,215,397,319)(183,281,368,216,398,320)(184,282,369,217,399,321)(185,283,370,218,400,322), (1,293,253,373)(2,294,254,374)(3,295,255,375)(4,296,256,376)(5,260,257,377)(6,261,258,378)(7,262,259,379)(8,263,223,380)(9,264,224,381)(10,265,225,382)(11,266,226,383)(12,267,227,384)(13,268,228,385)(14,269,229,386)(15,270,230,387)(16,271,231,388)(17,272,232,389)(18,273,233,390)(19,274,234,391)(20,275,235,392)(21,276,236,393)(22,277,237,394)(23,278,238,395)(24,279,239,396)(25,280,240,397)(26,281,241,398)(27,282,242,399)(28,283,243,400)(29,284,244,401)(30,285,245,402)(31,286,246,403)(32,287,247,404)(33,288,248,405)(34,289,249,406)(35,290,250,407)(36,291,251,371)(37,292,252,372)(38,184,88,217)(39,185,89,218)(40,149,90,219)(41,150,91,220)(42,151,92,221)(43,152,93,222)(44,153,94,186)(45,154,95,187)(46,155,96,188)(47,156,97,189)(48,157,98,190)(49,158,99,191)(50,159,100,192)(51,160,101,193)(52,161,102,194)(53,162,103,195)(54,163,104,196)(55,164,105,197)(56,165,106,198)(57,166,107,199)(58,167,108,200)(59,168,109,201)(60,169,110,202)(61,170,111,203)(62,171,75,204)(63,172,76,205)(64,173,77,206)(65,174,78,207)(66,175,79,208)(67,176,80,209)(68,177,81,210)(69,178,82,211)(70,179,83,212)(71,180,84,213)(72,181,85,214)(73,182,86,215)(74,183,87,216)(112,370,439,322)(113,334,440,323)(114,335,441,324)(115,336,442,325)(116,337,443,326)(117,338,444,327)(118,339,408,328)(119,340,409,329)(120,341,410,330)(121,342,411,331)(122,343,412,332)(123,344,413,333)(124,345,414,297)(125,346,415,298)(126,347,416,299)(127,348,417,300)(128,349,418,301)(129,350,419,302)(130,351,420,303)(131,352,421,304)(132,353,422,305)(133,354,423,306)(134,355,424,307)(135,356,425,308)(136,357,426,309)(137,358,427,310)(138,359,428,311)(139,360,429,312)(140,361,430,313)(141,362,431,314)(142,363,432,315)(143,364,433,316)(144,365,434,317)(145,366,435,318)(146,367,436,319)(147,368,437,320)(148,369,438,321) );
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),(334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370),(371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407),(408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444)], [(1,49,412,253,99,122),(2,50,413,254,100,123),(3,51,414,255,101,124),(4,52,415,256,102,125),(5,53,416,257,103,126),(6,54,417,258,104,127),(7,55,418,259,105,128),(8,56,419,223,106,129),(9,57,420,224,107,130),(10,58,421,225,108,131),(11,59,422,226,109,132),(12,60,423,227,110,133),(13,61,424,228,111,134),(14,62,425,229,75,135),(15,63,426,230,76,136),(16,64,427,231,77,137),(17,65,428,232,78,138),(18,66,429,233,79,139),(19,67,430,234,80,140),(20,68,431,235,81,141),(21,69,432,236,82,142),(22,70,433,237,83,143),(23,71,434,238,84,144),(24,72,435,239,85,145),(25,73,436,240,86,146),(26,74,437,241,87,147),(27,38,438,242,88,148),(28,39,439,243,89,112),(29,40,440,244,90,113),(30,41,441,245,91,114),(31,42,442,246,92,115),(32,43,443,247,93,116),(33,44,444,248,94,117),(34,45,408,249,95,118),(35,46,409,250,96,119),(36,47,410,251,97,120),(37,48,411,252,98,121),(149,284,334,219,401,323),(150,285,335,220,402,324),(151,286,336,221,403,325),(152,287,337,222,404,326),(153,288,338,186,405,327),(154,289,339,187,406,328),(155,290,340,188,407,329),(156,291,341,189,371,330),(157,292,342,190,372,331),(158,293,343,191,373,332),(159,294,344,192,374,333),(160,295,345,193,375,297),(161,296,346,194,376,298),(162,260,347,195,377,299),(163,261,348,196,378,300),(164,262,349,197,379,301),(165,263,350,198,380,302),(166,264,351,199,381,303),(167,265,352,200,382,304),(168,266,353,201,383,305),(169,267,354,202,384,306),(170,268,355,203,385,307),(171,269,356,204,386,308),(172,270,357,205,387,309),(173,271,358,206,388,310),(174,272,359,207,389,311),(175,273,360,208,390,312),(176,274,361,209,391,313),(177,275,362,210,392,314),(178,276,363,211,393,315),(179,277,364,212,394,316),(180,278,365,213,395,317),(181,279,366,214,396,318),(182,280,367,215,397,319),(183,281,368,216,398,320),(184,282,369,217,399,321),(185,283,370,218,400,322)], [(1,293,253,373),(2,294,254,374),(3,295,255,375),(4,296,256,376),(5,260,257,377),(6,261,258,378),(7,262,259,379),(8,263,223,380),(9,264,224,381),(10,265,225,382),(11,266,226,383),(12,267,227,384),(13,268,228,385),(14,269,229,386),(15,270,230,387),(16,271,231,388),(17,272,232,389),(18,273,233,390),(19,274,234,391),(20,275,235,392),(21,276,236,393),(22,277,237,394),(23,278,238,395),(24,279,239,396),(25,280,240,397),(26,281,241,398),(27,282,242,399),(28,283,243,400),(29,284,244,401),(30,285,245,402),(31,286,246,403),(32,287,247,404),(33,288,248,405),(34,289,249,406),(35,290,250,407),(36,291,251,371),(37,292,252,372),(38,184,88,217),(39,185,89,218),(40,149,90,219),(41,150,91,220),(42,151,92,221),(43,152,93,222),(44,153,94,186),(45,154,95,187),(46,155,96,188),(47,156,97,189),(48,157,98,190),(49,158,99,191),(50,159,100,192),(51,160,101,193),(52,161,102,194),(53,162,103,195),(54,163,104,196),(55,164,105,197),(56,165,106,198),(57,166,107,199),(58,167,108,200),(59,168,109,201),(60,169,110,202),(61,170,111,203),(62,171,75,204),(63,172,76,205),(64,173,77,206),(65,174,78,207),(66,175,79,208),(67,176,80,209),(68,177,81,210),(69,178,82,211),(70,179,83,212),(71,180,84,213),(72,181,85,214),(73,182,86,215),(74,183,87,216),(112,370,439,322),(113,334,440,323),(114,335,441,324),(115,336,442,325),(116,337,443,326),(117,338,444,327),(118,339,408,328),(119,340,409,329),(120,341,410,330),(121,342,411,331),(122,343,412,332),(123,344,413,333),(124,345,414,297),(125,346,415,298),(126,347,416,299),(127,348,417,300),(128,349,418,301),(129,350,419,302),(130,351,420,303),(131,352,421,304),(132,353,422,305),(133,354,423,306),(134,355,424,307),(135,356,425,308),(136,357,426,309),(137,358,427,310),(138,359,428,311),(139,360,429,312),(140,361,430,313),(141,362,431,314),(142,363,432,315),(143,364,433,316),(144,365,434,317),(145,366,435,318),(146,367,436,319),(147,368,437,320),(148,369,438,321)]])
222 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 37A | ··· | 37AJ | 74A | ··· | 74AJ | 111A | ··· | 111AJ | 148A | ··· | 148BT | 222A | ··· | 222AJ |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 37 | ··· | 37 | 74 | ··· | 74 | 111 | ··· | 111 | 148 | ··· | 148 | 222 | ··· | 222 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
222 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C37 | C74 | C148 | S3 | Dic3 | S3×C37 | Dic3×C37 |
| kernel | Dic3×C37 | C222 | C111 | Dic3 | C6 | C3 | C74 | C37 | C2 | C1 |
| # reps | 1 | 1 | 2 | 36 | 36 | 72 | 1 | 1 | 36 | 36 |
Matrix representation of Dic3×C37 ►in GL2(𝔽1777) generated by
| 693 | 0 |
| 0 | 693 |
| 1 | 1776 |
| 1 | 0 |
| 1529 | 601 |
| 353 | 248 |
G:=sub<GL(2,GF(1777))| [693,0,0,693],[1,1,1776,0],[1529,353,601,248] >;
Dic3×C37 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{37} % in TeX
G:=Group("Dic3xC37"); // GroupNames label
G:=SmallGroup(444,3);
// by ID
G=gap.SmallGroup(444,3);
# by ID
G:=PCGroup([4,-2,-37,-2,-3,296,4739]);
// Polycyclic
G:=Group<a,b,c|a^37=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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