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G = C3×Dic37order 444 = 22·3·37

Direct product of C3 and Dic37

Aliases: C3×Dic37, C375C12, C1114C4, C74.3C6, C6.2D37, C222.2C2, C2.(C3×D37), SmallGroup(444,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C37 — C3×Dic37
 Chief series C1 — C37 — C74 — C222 — C3×Dic37
 Lower central C37 — C3×Dic37
 Upper central C1 — C6

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
(1 213 87)(2 214 88)(3 215 89)(4 216 90)(5 217 91)(6 218 92)(7 219 93)(8 220 94)(9 221 95)(10 222 96)(11 149 97)(12 150 98)(13 151 99)(14 152 100)(15 153 101)(16 154 102)(17 155 103)(18 156 104)(19 157 105)(20 158 106)(21 159 107)(22 160 108)(23 161 109)(24 162 110)(25 163 111)(26 164 112)(27 165 113)(28 166 114)(29 167 115)(30 168 116)(31 169 117)(32 170 118)(33 171 119)(34 172 120)(35 173 121)(36 174 122)(37 175 123)(38 176 124)(39 177 125)(40 178 126)(41 179 127)(42 180 128)(43 181 129)(44 182 130)(45 183 131)(46 184 132)(47 185 133)(48 186 134)(49 187 135)(50 188 136)(51 189 137)(52 190 138)(53 191 139)(54 192 140)(55 193 141)(56 194 142)(57 195 143)(58 196 144)(59 197 145)(60 198 146)(61 199 147)(62 200 148)(63 201 75)(64 202 76)(65 203 77)(66 204 78)(67 205 79)(68 206 80)(69 207 81)(70 208 82)(71 209 83)(72 210 84)(73 211 85)(74 212 86)(223 408 297)(224 409 298)(225 410 299)(226 411 300)(227 412 301)(228 413 302)(229 414 303)(230 415 304)(231 416 305)(232 417 306)(233 418 307)(234 419 308)(235 420 309)(236 421 310)(237 422 311)(238 423 312)(239 424 313)(240 425 314)(241 426 315)(242 427 316)(243 428 317)(244 429 318)(245 430 319)(246 431 320)(247 432 321)(248 433 322)(249 434 323)(250 435 324)(251 436 325)(252 437 326)(253 438 327)(254 439 328)(255 440 329)(256 441 330)(257 442 331)(258 443 332)(259 444 333)(260 371 334)(261 372 335)(262 373 336)(263 374 337)(264 375 338)(265 376 339)(266 377 340)(267 378 341)(268 379 342)(269 380 343)(270 381 344)(271 382 345)(272 383 346)(273 384 347)(274 385 348)(275 386 349)(276 387 350)(277 388 351)(278 389 352)(279 390 353)(280 391 354)(281 392 355)(282 393 356)(283 394 357)(284 395 358)(285 396 359)(286 397 360)(287 398 361)(288 399 362)(289 400 363)(290 401 364)(291 402 365)(292 403 366)(293 404 367)(294 405 368)(295 406 369)(296 407 370)
(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)(75 309 112 346)(76 308 113 345)(77 307 114 344)(78 306 115 343)(79 305 116 342)(80 304 117 341)(81 303 118 340)(82 302 119 339)(83 301 120 338)(84 300 121 337)(85 299 122 336)(86 298 123 335)(87 297 124 334)(88 370 125 333)(89 369 126 332)(90 368 127 331)(91 367 128 330)(92 366 129 329)(93 365 130 328)(94 364 131 327)(95 363 132 326)(96 362 133 325)(97 361 134 324)(98 360 135 323)(99 359 136 322)(100 358 137 321)(101 357 138 320)(102 356 139 319)(103 355 140 318)(104 354 141 317)(105 353 142 316)(106 352 143 315)(107 351 144 314)(108 350 145 313)(109 349 146 312)(110 348 147 311)(111 347 148 310)(149 398 186 435)(150 397 187 434)(151 396 188 433)(152 395 189 432)(153 394 190 431)(154 393 191 430)(155 392 192 429)(156 391 193 428)(157 390 194 427)(158 389 195 426)(159 388 196 425)(160 387 197 424)(161 386 198 423)(162 385 199 422)(163 384 200 421)(164 383 201 420)(165 382 202 419)(166 381 203 418)(167 380 204 417)(168 379 205 416)(169 378 206 415)(170 377 207 414)(171 376 208 413)(172 375 209 412)(173 374 210 411)(174 373 211 410)(175 372 212 409)(176 371 213 408)(177 444 214 407)(178 443 215 406)(179 442 216 405)(180 441 217 404)(181 440 218 403)(182 439 219 402)(183 438 220 401)(184 437 221 400)(185 436 222 399)

G:=sub<Sym(444)| (1,213,87)(2,214,88)(3,215,89)(4,216,90)(5,217,91)(6,218,92)(7,219,93)(8,220,94)(9,221,95)(10,222,96)(11,149,97)(12,150,98)(13,151,99)(14,152,100)(15,153,101)(16,154,102)(17,155,103)(18,156,104)(19,157,105)(20,158,106)(21,159,107)(22,160,108)(23,161,109)(24,162,110)(25,163,111)(26,164,112)(27,165,113)(28,166,114)(29,167,115)(30,168,116)(31,169,117)(32,170,118)(33,171,119)(34,172,120)(35,173,121)(36,174,122)(37,175,123)(38,176,124)(39,177,125)(40,178,126)(41,179,127)(42,180,128)(43,181,129)(44,182,130)(45,183,131)(46,184,132)(47,185,133)(48,186,134)(49,187,135)(50,188,136)(51,189,137)(52,190,138)(53,191,139)(54,192,140)(55,193,141)(56,194,142)(57,195,143)(58,196,144)(59,197,145)(60,198,146)(61,199,147)(62,200,148)(63,201,75)(64,202,76)(65,203,77)(66,204,78)(67,205,79)(68,206,80)(69,207,81)(70,208,82)(71,209,83)(72,210,84)(73,211,85)(74,212,86)(223,408,297)(224,409,298)(225,410,299)(226,411,300)(227,412,301)(228,413,302)(229,414,303)(230,415,304)(231,416,305)(232,417,306)(233,418,307)(234,419,308)(235,420,309)(236,421,310)(237,422,311)(238,423,312)(239,424,313)(240,425,314)(241,426,315)(242,427,316)(243,428,317)(244,429,318)(245,430,319)(246,431,320)(247,432,321)(248,433,322)(249,434,323)(250,435,324)(251,436,325)(252,437,326)(253,438,327)(254,439,328)(255,440,329)(256,441,330)(257,442,331)(258,443,332)(259,444,333)(260,371,334)(261,372,335)(262,373,336)(263,374,337)(264,375,338)(265,376,339)(266,377,340)(267,378,341)(268,379,342)(269,380,343)(270,381,344)(271,382,345)(272,383,346)(273,384,347)(274,385,348)(275,386,349)(276,387,350)(277,388,351)(278,389,352)(279,390,353)(280,391,354)(281,392,355)(282,393,356)(283,394,357)(284,395,358)(285,396,359)(286,397,360)(287,398,361)(288,399,362)(289,400,363)(290,401,364)(291,402,365)(292,403,366)(293,404,367)(294,405,368)(295,406,369)(296,407,370), (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)(75,309,112,346)(76,308,113,345)(77,307,114,344)(78,306,115,343)(79,305,116,342)(80,304,117,341)(81,303,118,340)(82,302,119,339)(83,301,120,338)(84,300,121,337)(85,299,122,336)(86,298,123,335)(87,297,124,334)(88,370,125,333)(89,369,126,332)(90,368,127,331)(91,367,128,330)(92,366,129,329)(93,365,130,328)(94,364,131,327)(95,363,132,326)(96,362,133,325)(97,361,134,324)(98,360,135,323)(99,359,136,322)(100,358,137,321)(101,357,138,320)(102,356,139,319)(103,355,140,318)(104,354,141,317)(105,353,142,316)(106,352,143,315)(107,351,144,314)(108,350,145,313)(109,349,146,312)(110,348,147,311)(111,347,148,310)(149,398,186,435)(150,397,187,434)(151,396,188,433)(152,395,189,432)(153,394,190,431)(154,393,191,430)(155,392,192,429)(156,391,193,428)(157,390,194,427)(158,389,195,426)(159,388,196,425)(160,387,197,424)(161,386,198,423)(162,385,199,422)(163,384,200,421)(164,383,201,420)(165,382,202,419)(166,381,203,418)(167,380,204,417)(168,379,205,416)(169,378,206,415)(170,377,207,414)(171,376,208,413)(172,375,209,412)(173,374,210,411)(174,373,211,410)(175,372,212,409)(176,371,213,408)(177,444,214,407)(178,443,215,406)(179,442,216,405)(180,441,217,404)(181,440,218,403)(182,439,219,402)(183,438,220,401)(184,437,221,400)(185,436,222,399)>;

G:=Group( (1,213,87)(2,214,88)(3,215,89)(4,216,90)(5,217,91)(6,218,92)(7,219,93)(8,220,94)(9,221,95)(10,222,96)(11,149,97)(12,150,98)(13,151,99)(14,152,100)(15,153,101)(16,154,102)(17,155,103)(18,156,104)(19,157,105)(20,158,106)(21,159,107)(22,160,108)(23,161,109)(24,162,110)(25,163,111)(26,164,112)(27,165,113)(28,166,114)(29,167,115)(30,168,116)(31,169,117)(32,170,118)(33,171,119)(34,172,120)(35,173,121)(36,174,122)(37,175,123)(38,176,124)(39,177,125)(40,178,126)(41,179,127)(42,180,128)(43,181,129)(44,182,130)(45,183,131)(46,184,132)(47,185,133)(48,186,134)(49,187,135)(50,188,136)(51,189,137)(52,190,138)(53,191,139)(54,192,140)(55,193,141)(56,194,142)(57,195,143)(58,196,144)(59,197,145)(60,198,146)(61,199,147)(62,200,148)(63,201,75)(64,202,76)(65,203,77)(66,204,78)(67,205,79)(68,206,80)(69,207,81)(70,208,82)(71,209,83)(72,210,84)(73,211,85)(74,212,86)(223,408,297)(224,409,298)(225,410,299)(226,411,300)(227,412,301)(228,413,302)(229,414,303)(230,415,304)(231,416,305)(232,417,306)(233,418,307)(234,419,308)(235,420,309)(236,421,310)(237,422,311)(238,423,312)(239,424,313)(240,425,314)(241,426,315)(242,427,316)(243,428,317)(244,429,318)(245,430,319)(246,431,320)(247,432,321)(248,433,322)(249,434,323)(250,435,324)(251,436,325)(252,437,326)(253,438,327)(254,439,328)(255,440,329)(256,441,330)(257,442,331)(258,443,332)(259,444,333)(260,371,334)(261,372,335)(262,373,336)(263,374,337)(264,375,338)(265,376,339)(266,377,340)(267,378,341)(268,379,342)(269,380,343)(270,381,344)(271,382,345)(272,383,346)(273,384,347)(274,385,348)(275,386,349)(276,387,350)(277,388,351)(278,389,352)(279,390,353)(280,391,354)(281,392,355)(282,393,356)(283,394,357)(284,395,358)(285,396,359)(286,397,360)(287,398,361)(288,399,362)(289,400,363)(290,401,364)(291,402,365)(292,403,366)(293,404,367)(294,405,368)(295,406,369)(296,407,370), (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)(75,309,112,346)(76,308,113,345)(77,307,114,344)(78,306,115,343)(79,305,116,342)(80,304,117,341)(81,303,118,340)(82,302,119,339)(83,301,120,338)(84,300,121,337)(85,299,122,336)(86,298,123,335)(87,297,124,334)(88,370,125,333)(89,369,126,332)(90,368,127,331)(91,367,128,330)(92,366,129,329)(93,365,130,328)(94,364,131,327)(95,363,132,326)(96,362,133,325)(97,361,134,324)(98,360,135,323)(99,359,136,322)(100,358,137,321)(101,357,138,320)(102,356,139,319)(103,355,140,318)(104,354,141,317)(105,353,142,316)(106,352,143,315)(107,351,144,314)(108,350,145,313)(109,349,146,312)(110,348,147,311)(111,347,148,310)(149,398,186,435)(150,397,187,434)(151,396,188,433)(152,395,189,432)(153,394,190,431)(154,393,191,430)(155,392,192,429)(156,391,193,428)(157,390,194,427)(158,389,195,426)(159,388,196,425)(160,387,197,424)(161,386,198,423)(162,385,199,422)(163,384,200,421)(164,383,201,420)(165,382,202,419)(166,381,203,418)(167,380,204,417)(168,379,205,416)(169,378,206,415)(170,377,207,414)(171,376,208,413)(172,375,209,412)(173,374,210,411)(174,373,211,410)(175,372,212,409)(176,371,213,408)(177,444,214,407)(178,443,215,406)(179,442,216,405)(180,441,217,404)(181,440,218,403)(182,439,219,402)(183,438,220,401)(184,437,221,400)(185,436,222,399) );

G=PermutationGroup([(1,213,87),(2,214,88),(3,215,89),(4,216,90),(5,217,91),(6,218,92),(7,219,93),(8,220,94),(9,221,95),(10,222,96),(11,149,97),(12,150,98),(13,151,99),(14,152,100),(15,153,101),(16,154,102),(17,155,103),(18,156,104),(19,157,105),(20,158,106),(21,159,107),(22,160,108),(23,161,109),(24,162,110),(25,163,111),(26,164,112),(27,165,113),(28,166,114),(29,167,115),(30,168,116),(31,169,117),(32,170,118),(33,171,119),(34,172,120),(35,173,121),(36,174,122),(37,175,123),(38,176,124),(39,177,125),(40,178,126),(41,179,127),(42,180,128),(43,181,129),(44,182,130),(45,183,131),(46,184,132),(47,185,133),(48,186,134),(49,187,135),(50,188,136),(51,189,137),(52,190,138),(53,191,139),(54,192,140),(55,193,141),(56,194,142),(57,195,143),(58,196,144),(59,197,145),(60,198,146),(61,199,147),(62,200,148),(63,201,75),(64,202,76),(65,203,77),(66,204,78),(67,205,79),(68,206,80),(69,207,81),(70,208,82),(71,209,83),(72,210,84),(73,211,85),(74,212,86),(223,408,297),(224,409,298),(225,410,299),(226,411,300),(227,412,301),(228,413,302),(229,414,303),(230,415,304),(231,416,305),(232,417,306),(233,418,307),(234,419,308),(235,420,309),(236,421,310),(237,422,311),(238,423,312),(239,424,313),(240,425,314),(241,426,315),(242,427,316),(243,428,317),(244,429,318),(245,430,319),(246,431,320),(247,432,321),(248,433,322),(249,434,323),(250,435,324),(251,436,325),(252,437,326),(253,438,327),(254,439,328),(255,440,329),(256,441,330),(257,442,331),(258,443,332),(259,444,333),(260,371,334),(261,372,335),(262,373,336),(263,374,337),(264,375,338),(265,376,339),(266,377,340),(267,378,341),(268,379,342),(269,380,343),(270,381,344),(271,382,345),(272,383,346),(273,384,347),(274,385,348),(275,386,349),(276,387,350),(277,388,351),(278,389,352),(279,390,353),(280,391,354),(281,392,355),(282,393,356),(283,394,357),(284,395,358),(285,396,359),(286,397,360),(287,398,361),(288,399,362),(289,400,363),(290,401,364),(291,402,365),(292,403,366),(293,404,367),(294,405,368),(295,406,369),(296,407,370)], [(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),(75,309,112,346),(76,308,113,345),(77,307,114,344),(78,306,115,343),(79,305,116,342),(80,304,117,341),(81,303,118,340),(82,302,119,339),(83,301,120,338),(84,300,121,337),(85,299,122,336),(86,298,123,335),(87,297,124,334),(88,370,125,333),(89,369,126,332),(90,368,127,331),(91,367,128,330),(92,366,129,329),(93,365,130,328),(94,364,131,327),(95,363,132,326),(96,362,133,325),(97,361,134,324),(98,360,135,323),(99,359,136,322),(100,358,137,321),(101,357,138,320),(102,356,139,319),(103,355,140,318),(104,354,141,317),(105,353,142,316),(106,352,143,315),(107,351,144,314),(108,350,145,313),(109,349,146,312),(110,348,147,311),(111,347,148,310),(149,398,186,435),(150,397,187,434),(151,396,188,433),(152,395,189,432),(153,394,190,431),(154,393,191,430),(155,392,192,429),(156,391,193,428),(157,390,194,427),(158,389,195,426),(159,388,196,425),(160,387,197,424),(161,386,198,423),(162,385,199,422),(163,384,200,421),(164,383,201,420),(165,382,202,419),(166,381,203,418),(167,380,204,417),(168,379,205,416),(169,378,206,415),(170,377,207,414),(171,376,208,413),(172,375,209,412),(173,374,210,411),(174,373,211,410),(175,372,212,409),(176,371,213,408),(177,444,214,407),(178,443,215,406),(179,442,216,405),(180,441,217,404),(181,440,218,403),(182,439,219,402),(183,438,220,401),(184,437,221,400),(185,436,222,399)])

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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