direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic37, C37⋊5C12, C111⋊4C4, C74.3C6, C6.2D37, C222.2C2, C2.(C3×D37), SmallGroup(444,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C37 — C3×Dic37 |
Generators and relations for C3×Dic37
G = < a,b,c | a3=b74=1, c2=b37, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic37
►Regular action on 444 points
Generators in S444
(1 154 80)(2 155 81)(3 156 82)(4 157 83)(5 158 84)(6 159 85)(7 160 86)(8 161 87)(9 162 88)(10 163 89)(11 164 90)(12 165 91)(13 166 92)(14 167 93)(15 168 94)(16 169 95)(17 170 96)(18 171 97)(19 172 98)(20 173 99)(21 174 100)(22 175 101)(23 176 102)(24 177 103)(25 178 104)(26 179 105)(27 180 106)(28 181 107)(29 182 108)(30 183 109)(31 184 110)(32 185 111)(33 186 112)(34 187 113)(35 188 114)(36 189 115)(37 190 116)(38 191 117)(39 192 118)(40 193 119)(41 194 120)(42 195 121)(43 196 122)(44 197 123)(45 198 124)(46 199 125)(47 200 126)(48 201 127)(49 202 128)(50 203 129)(51 204 130)(52 205 131)(53 206 132)(54 207 133)(55 208 134)(56 209 135)(57 210 136)(58 211 137)(59 212 138)(60 213 139)(61 214 140)(62 215 141)(63 216 142)(64 217 143)(65 218 144)(66 219 145)(67 220 146)(68 221 147)(69 222 148)(70 149 75)(71 150 76)(72 151 77)(73 152 78)(74 153 79)(223 371 334)(224 372 335)(225 373 336)(226 374 337)(227 375 338)(228 376 339)(229 377 340)(230 378 341)(231 379 342)(232 380 343)(233 381 344)(234 382 345)(235 383 346)(236 384 347)(237 385 348)(238 386 349)(239 387 350)(240 388 351)(241 389 352)(242 390 353)(243 391 354)(244 392 355)(245 393 356)(246 394 357)(247 395 358)(248 396 359)(249 397 360)(250 398 361)(251 399 362)(252 400 363)(253 401 364)(254 402 365)(255 403 366)(256 404 367)(257 405 368)(258 406 369)(259 407 370)(260 408 297)(261 409 298)(262 410 299)(263 411 300)(264 412 301)(265 413 302)(266 414 303)(267 415 304)(268 416 305)(269 417 306)(270 418 307)(271 419 308)(272 420 309)(273 421 310)(274 422 311)(275 423 312)(276 424 313)(277 425 314)(278 426 315)(279 427 316)(280 428 317)(281 429 318)(282 430 319)(283 431 320)(284 432 321)(285 433 322)(286 434 323)(287 435 324)(288 436 325)(289 437 326)(290 438 327)(291 439 328)(292 440 329)(293 441 330)(294 442 331)(295 443 332)(296 444 333)
(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)
(1 223 38 260)(2 296 39 259)(3 295 40 258)(4 294 41 257)(5 293 42 256)(6 292 43 255)(7 291 44 254)(8 290 45 253)(9 289 46 252)(10 288 47 251)(11 287 48 250)(12 286 49 249)(13 285 50 248)(14 284 51 247)(15 283 52 246)(16 282 53 245)(17 281 54 244)(18 280 55 243)(19 279 56 242)(20 278 57 241)(21 277 58 240)(22 276 59 239)(23 275 60 238)(24 274 61 237)(25 273 62 236)(26 272 63 235)(27 271 64 234)(28 270 65 233)(29 269 66 232)(30 268 67 231)(31 267 68 230)(32 266 69 229)(33 265 70 228)(34 264 71 227)(35 263 72 226)(36 262 73 225)(37 261 74 224)(75 339 112 302)(76 338 113 301)(77 337 114 300)(78 336 115 299)(79 335 116 298)(80 334 117 297)(81 333 118 370)(82 332 119 369)(83 331 120 368)(84 330 121 367)(85 329 122 366)(86 328 123 365)(87 327 124 364)(88 326 125 363)(89 325 126 362)(90 324 127 361)(91 323 128 360)(92 322 129 359)(93 321 130 358)(94 320 131 357)(95 319 132 356)(96 318 133 355)(97 317 134 354)(98 316 135 353)(99 315 136 352)(100 314 137 351)(101 313 138 350)(102 312 139 349)(103 311 140 348)(104 310 141 347)(105 309 142 346)(106 308 143 345)(107 307 144 344)(108 306 145 343)(109 305 146 342)(110 304 147 341)(111 303 148 340)(149 376 186 413)(150 375 187 412)(151 374 188 411)(152 373 189 410)(153 372 190 409)(154 371 191 408)(155 444 192 407)(156 443 193 406)(157 442 194 405)(158 441 195 404)(159 440 196 403)(160 439 197 402)(161 438 198 401)(162 437 199 400)(163 436 200 399)(164 435 201 398)(165 434 202 397)(166 433 203 396)(167 432 204 395)(168 431 205 394)(169 430 206 393)(170 429 207 392)(171 428 208 391)(172 427 209 390)(173 426 210 389)(174 425 211 388)(175 424 212 387)(176 423 213 386)(177 422 214 385)(178 421 215 384)(179 420 216 383)(180 419 217 382)(181 418 218 381)(182 417 219 380)(183 416 220 379)(184 415 221 378)(185 414 222 377)
G:=sub<Sym(444)| (1,154,80)(2,155,81)(3,156,82)(4,157,83)(5,158,84)(6,159,85)(7,160,86)(8,161,87)(9,162,88)(10,163,89)(11,164,90)(12,165,91)(13,166,92)(14,167,93)(15,168,94)(16,169,95)(17,170,96)(18,171,97)(19,172,98)(20,173,99)(21,174,100)(22,175,101)(23,176,102)(24,177,103)(25,178,104)(26,179,105)(27,180,106)(28,181,107)(29,182,108)(30,183,109)(31,184,110)(32,185,111)(33,186,112)(34,187,113)(35,188,114)(36,189,115)(37,190,116)(38,191,117)(39,192,118)(40,193,119)(41,194,120)(42,195,121)(43,196,122)(44,197,123)(45,198,124)(46,199,125)(47,200,126)(48,201,127)(49,202,128)(50,203,129)(51,204,130)(52,205,131)(53,206,132)(54,207,133)(55,208,134)(56,209,135)(57,210,136)(58,211,137)(59,212,138)(60,213,139)(61,214,140)(62,215,141)(63,216,142)(64,217,143)(65,218,144)(66,219,145)(67,220,146)(68,221,147)(69,222,148)(70,149,75)(71,150,76)(72,151,77)(73,152,78)(74,153,79)(223,371,334)(224,372,335)(225,373,336)(226,374,337)(227,375,338)(228,376,339)(229,377,340)(230,378,341)(231,379,342)(232,380,343)(233,381,344)(234,382,345)(235,383,346)(236,384,347)(237,385,348)(238,386,349)(239,387,350)(240,388,351)(241,389,352)(242,390,353)(243,391,354)(244,392,355)(245,393,356)(246,394,357)(247,395,358)(248,396,359)(249,397,360)(250,398,361)(251,399,362)(252,400,363)(253,401,364)(254,402,365)(255,403,366)(256,404,367)(257,405,368)(258,406,369)(259,407,370)(260,408,297)(261,409,298)(262,410,299)(263,411,300)(264,412,301)(265,413,302)(266,414,303)(267,415,304)(268,416,305)(269,417,306)(270,418,307)(271,419,308)(272,420,309)(273,421,310)(274,422,311)(275,423,312)(276,424,313)(277,425,314)(278,426,315)(279,427,316)(280,428,317)(281,429,318)(282,430,319)(283,431,320)(284,432,321)(285,433,322)(286,434,323)(287,435,324)(288,436,325)(289,437,326)(290,438,327)(291,439,328)(292,440,329)(293,441,330)(294,442,331)(295,443,332)(296,444,333), (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), (1,223,38,260)(2,296,39,259)(3,295,40,258)(4,294,41,257)(5,293,42,256)(6,292,43,255)(7,291,44,254)(8,290,45,253)(9,289,46,252)(10,288,47,251)(11,287,48,250)(12,286,49,249)(13,285,50,248)(14,284,51,247)(15,283,52,246)(16,282,53,245)(17,281,54,244)(18,280,55,243)(19,279,56,242)(20,278,57,241)(21,277,58,240)(22,276,59,239)(23,275,60,238)(24,274,61,237)(25,273,62,236)(26,272,63,235)(27,271,64,234)(28,270,65,233)(29,269,66,232)(30,268,67,231)(31,267,68,230)(32,266,69,229)(33,265,70,228)(34,264,71,227)(35,263,72,226)(36,262,73,225)(37,261,74,224)(75,339,112,302)(76,338,113,301)(77,337,114,300)(78,336,115,299)(79,335,116,298)(80,334,117,297)(81,333,118,370)(82,332,119,369)(83,331,120,368)(84,330,121,367)(85,329,122,366)(86,328,123,365)(87,327,124,364)(88,326,125,363)(89,325,126,362)(90,324,127,361)(91,323,128,360)(92,322,129,359)(93,321,130,358)(94,320,131,357)(95,319,132,356)(96,318,133,355)(97,317,134,354)(98,316,135,353)(99,315,136,352)(100,314,137,351)(101,313,138,350)(102,312,139,349)(103,311,140,348)(104,310,141,347)(105,309,142,346)(106,308,143,345)(107,307,144,344)(108,306,145,343)(109,305,146,342)(110,304,147,341)(111,303,148,340)(149,376,186,413)(150,375,187,412)(151,374,188,411)(152,373,189,410)(153,372,190,409)(154,371,191,408)(155,444,192,407)(156,443,193,406)(157,442,194,405)(158,441,195,404)(159,440,196,403)(160,439,197,402)(161,438,198,401)(162,437,199,400)(163,436,200,399)(164,435,201,398)(165,434,202,397)(166,433,203,396)(167,432,204,395)(168,431,205,394)(169,430,206,393)(170,429,207,392)(171,428,208,391)(172,427,209,390)(173,426,210,389)(174,425,211,388)(175,424,212,387)(176,423,213,386)(177,422,214,385)(178,421,215,384)(179,420,216,383)(180,419,217,382)(181,418,218,381)(182,417,219,380)(183,416,220,379)(184,415,221,378)(185,414,222,377)>;
G:=Group( (1,154,80)(2,155,81)(3,156,82)(4,157,83)(5,158,84)(6,159,85)(7,160,86)(8,161,87)(9,162,88)(10,163,89)(11,164,90)(12,165,91)(13,166,92)(14,167,93)(15,168,94)(16,169,95)(17,170,96)(18,171,97)(19,172,98)(20,173,99)(21,174,100)(22,175,101)(23,176,102)(24,177,103)(25,178,104)(26,179,105)(27,180,106)(28,181,107)(29,182,108)(30,183,109)(31,184,110)(32,185,111)(33,186,112)(34,187,113)(35,188,114)(36,189,115)(37,190,116)(38,191,117)(39,192,118)(40,193,119)(41,194,120)(42,195,121)(43,196,122)(44,197,123)(45,198,124)(46,199,125)(47,200,126)(48,201,127)(49,202,128)(50,203,129)(51,204,130)(52,205,131)(53,206,132)(54,207,133)(55,208,134)(56,209,135)(57,210,136)(58,211,137)(59,212,138)(60,213,139)(61,214,140)(62,215,141)(63,216,142)(64,217,143)(65,218,144)(66,219,145)(67,220,146)(68,221,147)(69,222,148)(70,149,75)(71,150,76)(72,151,77)(73,152,78)(74,153,79)(223,371,334)(224,372,335)(225,373,336)(226,374,337)(227,375,338)(228,376,339)(229,377,340)(230,378,341)(231,379,342)(232,380,343)(233,381,344)(234,382,345)(235,383,346)(236,384,347)(237,385,348)(238,386,349)(239,387,350)(240,388,351)(241,389,352)(242,390,353)(243,391,354)(244,392,355)(245,393,356)(246,394,357)(247,395,358)(248,396,359)(249,397,360)(250,398,361)(251,399,362)(252,400,363)(253,401,364)(254,402,365)(255,403,366)(256,404,367)(257,405,368)(258,406,369)(259,407,370)(260,408,297)(261,409,298)(262,410,299)(263,411,300)(264,412,301)(265,413,302)(266,414,303)(267,415,304)(268,416,305)(269,417,306)(270,418,307)(271,419,308)(272,420,309)(273,421,310)(274,422,311)(275,423,312)(276,424,313)(277,425,314)(278,426,315)(279,427,316)(280,428,317)(281,429,318)(282,430,319)(283,431,320)(284,432,321)(285,433,322)(286,434,323)(287,435,324)(288,436,325)(289,437,326)(290,438,327)(291,439,328)(292,440,329)(293,441,330)(294,442,331)(295,443,332)(296,444,333), (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), (1,223,38,260)(2,296,39,259)(3,295,40,258)(4,294,41,257)(5,293,42,256)(6,292,43,255)(7,291,44,254)(8,290,45,253)(9,289,46,252)(10,288,47,251)(11,287,48,250)(12,286,49,249)(13,285,50,248)(14,284,51,247)(15,283,52,246)(16,282,53,245)(17,281,54,244)(18,280,55,243)(19,279,56,242)(20,278,57,241)(21,277,58,240)(22,276,59,239)(23,275,60,238)(24,274,61,237)(25,273,62,236)(26,272,63,235)(27,271,64,234)(28,270,65,233)(29,269,66,232)(30,268,67,231)(31,267,68,230)(32,266,69,229)(33,265,70,228)(34,264,71,227)(35,263,72,226)(36,262,73,225)(37,261,74,224)(75,339,112,302)(76,338,113,301)(77,337,114,300)(78,336,115,299)(79,335,116,298)(80,334,117,297)(81,333,118,370)(82,332,119,369)(83,331,120,368)(84,330,121,367)(85,329,122,366)(86,328,123,365)(87,327,124,364)(88,326,125,363)(89,325,126,362)(90,324,127,361)(91,323,128,360)(92,322,129,359)(93,321,130,358)(94,320,131,357)(95,319,132,356)(96,318,133,355)(97,317,134,354)(98,316,135,353)(99,315,136,352)(100,314,137,351)(101,313,138,350)(102,312,139,349)(103,311,140,348)(104,310,141,347)(105,309,142,346)(106,308,143,345)(107,307,144,344)(108,306,145,343)(109,305,146,342)(110,304,147,341)(111,303,148,340)(149,376,186,413)(150,375,187,412)(151,374,188,411)(152,373,189,410)(153,372,190,409)(154,371,191,408)(155,444,192,407)(156,443,193,406)(157,442,194,405)(158,441,195,404)(159,440,196,403)(160,439,197,402)(161,438,198,401)(162,437,199,400)(163,436,200,399)(164,435,201,398)(165,434,202,397)(166,433,203,396)(167,432,204,395)(168,431,205,394)(169,430,206,393)(170,429,207,392)(171,428,208,391)(172,427,209,390)(173,426,210,389)(174,425,211,388)(175,424,212,387)(176,423,213,386)(177,422,214,385)(178,421,215,384)(179,420,216,383)(180,419,217,382)(181,418,218,381)(182,417,219,380)(183,416,220,379)(184,415,221,378)(185,414,222,377) );
G=PermutationGroup([[(1,154,80),(2,155,81),(3,156,82),(4,157,83),(5,158,84),(6,159,85),(7,160,86),(8,161,87),(9,162,88),(10,163,89),(11,164,90),(12,165,91),(13,166,92),(14,167,93),(15,168,94),(16,169,95),(17,170,96),(18,171,97),(19,172,98),(20,173,99),(21,174,100),(22,175,101),(23,176,102),(24,177,103),(25,178,104),(26,179,105),(27,180,106),(28,181,107),(29,182,108),(30,183,109),(31,184,110),(32,185,111),(33,186,112),(34,187,113),(35,188,114),(36,189,115),(37,190,116),(38,191,117),(39,192,118),(40,193,119),(41,194,120),(42,195,121),(43,196,122),(44,197,123),(45,198,124),(46,199,125),(47,200,126),(48,201,127),(49,202,128),(50,203,129),(51,204,130),(52,205,131),(53,206,132),(54,207,133),(55,208,134),(56,209,135),(57,210,136),(58,211,137),(59,212,138),(60,213,139),(61,214,140),(62,215,141),(63,216,142),(64,217,143),(65,218,144),(66,219,145),(67,220,146),(68,221,147),(69,222,148),(70,149,75),(71,150,76),(72,151,77),(73,152,78),(74,153,79),(223,371,334),(224,372,335),(225,373,336),(226,374,337),(227,375,338),(228,376,339),(229,377,340),(230,378,341),(231,379,342),(232,380,343),(233,381,344),(234,382,345),(235,383,346),(236,384,347),(237,385,348),(238,386,349),(239,387,350),(240,388,351),(241,389,352),(242,390,353),(243,391,354),(244,392,355),(245,393,356),(246,394,357),(247,395,358),(248,396,359),(249,397,360),(250,398,361),(251,399,362),(252,400,363),(253,401,364),(254,402,365),(255,403,366),(256,404,367),(257,405,368),(258,406,369),(259,407,370),(260,408,297),(261,409,298),(262,410,299),(263,411,300),(264,412,301),(265,413,302),(266,414,303),(267,415,304),(268,416,305),(269,417,306),(270,418,307),(271,419,308),(272,420,309),(273,421,310),(274,422,311),(275,423,312),(276,424,313),(277,425,314),(278,426,315),(279,427,316),(280,428,317),(281,429,318),(282,430,319),(283,431,320),(284,432,321),(285,433,322),(286,434,323),(287,435,324),(288,436,325),(289,437,326),(290,438,327),(291,439,328),(292,440,329),(293,441,330),(294,442,331),(295,443,332),(296,444,333)], [(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)], [(1,223,38,260),(2,296,39,259),(3,295,40,258),(4,294,41,257),(5,293,42,256),(6,292,43,255),(7,291,44,254),(8,290,45,253),(9,289,46,252),(10,288,47,251),(11,287,48,250),(12,286,49,249),(13,285,50,248),(14,284,51,247),(15,283,52,246),(16,282,53,245),(17,281,54,244),(18,280,55,243),(19,279,56,242),(20,278,57,241),(21,277,58,240),(22,276,59,239),(23,275,60,238),(24,274,61,237),(25,273,62,236),(26,272,63,235),(27,271,64,234),(28,270,65,233),(29,269,66,232),(30,268,67,231),(31,267,68,230),(32,266,69,229),(33,265,70,228),(34,264,71,227),(35,263,72,226),(36,262,73,225),(37,261,74,224),(75,339,112,302),(76,338,113,301),(77,337,114,300),(78,336,115,299),(79,335,116,298),(80,334,117,297),(81,333,118,370),(82,332,119,369),(83,331,120,368),(84,330,121,367),(85,329,122,366),(86,328,123,365),(87,327,124,364),(88,326,125,363),(89,325,126,362),(90,324,127,361),(91,323,128,360),(92,322,129,359),(93,321,130,358),(94,320,131,357),(95,319,132,356),(96,318,133,355),(97,317,134,354),(98,316,135,353),(99,315,136,352),(100,314,137,351),(101,313,138,350),(102,312,139,349),(103,311,140,348),(104,310,141,347),(105,309,142,346),(106,308,143,345),(107,307,144,344),(108,306,145,343),(109,305,146,342),(110,304,147,341),(111,303,148,340),(149,376,186,413),(150,375,187,412),(151,374,188,411),(152,373,189,410),(153,372,190,409),(154,371,191,408),(155,444,192,407),(156,443,193,406),(157,442,194,405),(158,441,195,404),(159,440,196,403),(160,439,197,402),(161,438,198,401),(162,437,199,400),(163,436,200,399),(164,435,201,398),(165,434,202,397),(166,433,203,396),(167,432,204,395),(168,431,205,394),(169,430,206,393),(170,429,207,392),(171,428,208,391),(172,427,209,390),(173,426,210,389),(174,425,211,388),(175,424,212,387),(176,423,213,386),(177,422,214,385),(178,421,215,384),(179,420,216,383),(180,419,217,382),(181,418,218,381),(182,417,219,380),(183,416,220,379),(184,415,221,378),(185,414,222,377)]])
120 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 37A | ··· | 37R | 74A | ··· | 74R | 111A | ··· | 111AJ | 222A | ··· | 222AJ |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 37 | ··· | 37 | 74 | ··· | 74 | 111 | ··· | 111 | 222 | ··· | 222 |
| size | 1 | 1 | 1 | 1 | 37 | 37 | 1 | 1 | 37 | 37 | 37 | 37 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D37 | Dic37 | C3×D37 | C3×Dic37 |
| kernel | C3×Dic37 | C222 | Dic37 | C111 | C74 | C37 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 18 | 18 | 36 | 36 |
Matrix representation of C3×Dic37 ►in GL2(𝔽1777) generated by
| 1147 | 0 |
| 0 | 1147 |
| 0 | 1776 |
| 1 | 198 |
| 148 | 1555 |
| 683 | 1629 |
G:=sub<GL(2,GF(1777))| [1147,0,0,1147],[0,1,1776,198],[148,683,1555,1629] >;
C3×Dic37 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{37} % in TeX
G:=Group("C3xDic37"); // GroupNames label
G:=SmallGroup(444,4);
// by ID
G=gap.SmallGroup(444,4);
# by ID
G:=PCGroup([4,-2,-3,-2,-37,24,6915]);
// Polycyclic
G:=Group<a,b,c|a^3=b^74=1,c^2=b^37,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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