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G = C42×C28order 448 = 26·7

Abelian group of type [4,4,28]

direct product, abelian, monomial, 2-elementary

Aliases: C42×C28, SmallGroup(448,782)

Series: Derived Chief Lower central Upper central

C1 — C42×C28
C1C2C22C23C22×C14C22×C28C2×C4×C28 — C42×C28
C1 — C42×C28
C1 — C42×C28

Generators and relations for C42×C28
 G = < a,b,c | a4=b4=c28=1, ab=ba, ac=ca, bc=cb >

Subgroups: 258, all normal (6 characteristic)
C1, C2 [×7], C4 [×28], C22 [×7], C7, C2×C4 [×42], C23, C14 [×7], C42 [×28], C22×C4 [×7], C28 [×28], C2×C14 [×7], C2×C42 [×7], C2×C28 [×42], C22×C14, C43, C4×C28 [×28], C22×C28 [×7], C2×C4×C28 [×7], C42×C28
Quotients: C1, C2 [×7], C4 [×28], C22 [×7], C7, C2×C4 [×42], C23, C14 [×7], C42 [×28], C22×C4 [×7], C28 [×28], C2×C14 [×7], C2×C42 [×7], C2×C28 [×42], C22×C14, C43, C4×C28 [×28], C22×C28 [×7], C2×C4×C28 [×7], C42×C28

Smallest permutation representation of C42×C28
Regular action on 448 points
Generators in S448
(1 277 163 86)(2 278 164 87)(3 279 165 88)(4 280 166 89)(5 253 167 90)(6 254 168 91)(7 255 141 92)(8 256 142 93)(9 257 143 94)(10 258 144 95)(11 259 145 96)(12 260 146 97)(13 261 147 98)(14 262 148 99)(15 263 149 100)(16 264 150 101)(17 265 151 102)(18 266 152 103)(19 267 153 104)(20 268 154 105)(21 269 155 106)(22 270 156 107)(23 271 157 108)(24 272 158 109)(25 273 159 110)(26 274 160 111)(27 275 161 112)(28 276 162 85)(29 342 234 399)(30 343 235 400)(31 344 236 401)(32 345 237 402)(33 346 238 403)(34 347 239 404)(35 348 240 405)(36 349 241 406)(37 350 242 407)(38 351 243 408)(39 352 244 409)(40 353 245 410)(41 354 246 411)(42 355 247 412)(43 356 248 413)(44 357 249 414)(45 358 250 415)(46 359 251 416)(47 360 252 417)(48 361 225 418)(49 362 226 419)(50 363 227 420)(51 364 228 393)(52 337 229 394)(53 338 230 395)(54 339 231 396)(55 340 232 397)(56 341 233 398)(57 324 209 370)(58 325 210 371)(59 326 211 372)(60 327 212 373)(61 328 213 374)(62 329 214 375)(63 330 215 376)(64 331 216 377)(65 332 217 378)(66 333 218 379)(67 334 219 380)(68 335 220 381)(69 336 221 382)(70 309 222 383)(71 310 223 384)(72 311 224 385)(73 312 197 386)(74 313 198 387)(75 314 199 388)(76 315 200 389)(77 316 201 390)(78 317 202 391)(79 318 203 392)(80 319 204 365)(81 320 205 366)(82 321 206 367)(83 322 207 368)(84 323 208 369)(113 288 184 429)(114 289 185 430)(115 290 186 431)(116 291 187 432)(117 292 188 433)(118 293 189 434)(119 294 190 435)(120 295 191 436)(121 296 192 437)(122 297 193 438)(123 298 194 439)(124 299 195 440)(125 300 196 441)(126 301 169 442)(127 302 170 443)(128 303 171 444)(129 304 172 445)(130 305 173 446)(131 306 174 447)(132 307 175 448)(133 308 176 421)(134 281 177 422)(135 282 178 423)(136 283 179 424)(137 284 180 425)(138 285 181 426)(139 286 182 427)(140 287 183 428)
(1 298 51 203)(2 299 52 204)(3 300 53 205)(4 301 54 206)(5 302 55 207)(6 303 56 208)(7 304 29 209)(8 305 30 210)(9 306 31 211)(10 307 32 212)(11 308 33 213)(12 281 34 214)(13 282 35 215)(14 283 36 216)(15 284 37 217)(16 285 38 218)(17 286 39 219)(18 287 40 220)(19 288 41 221)(20 289 42 222)(21 290 43 223)(22 291 44 224)(23 292 45 197)(24 293 46 198)(25 294 47 199)(26 295 48 200)(27 296 49 201)(28 297 50 202)(57 141 445 234)(58 142 446 235)(59 143 447 236)(60 144 448 237)(61 145 421 238)(62 146 422 239)(63 147 423 240)(64 148 424 241)(65 149 425 242)(66 150 426 243)(67 151 427 244)(68 152 428 245)(69 153 429 246)(70 154 430 247)(71 155 431 248)(72 156 432 249)(73 157 433 250)(74 158 434 251)(75 159 435 252)(76 160 436 225)(77 161 437 226)(78 162 438 227)(79 163 439 228)(80 164 440 229)(81 165 441 230)(82 166 442 231)(83 167 443 232)(84 168 444 233)(85 122 420 317)(86 123 393 318)(87 124 394 319)(88 125 395 320)(89 126 396 321)(90 127 397 322)(91 128 398 323)(92 129 399 324)(93 130 400 325)(94 131 401 326)(95 132 402 327)(96 133 403 328)(97 134 404 329)(98 135 405 330)(99 136 406 331)(100 137 407 332)(101 138 408 333)(102 139 409 334)(103 140 410 335)(104 113 411 336)(105 114 412 309)(106 115 413 310)(107 116 414 311)(108 117 415 312)(109 118 416 313)(110 119 417 314)(111 120 418 315)(112 121 419 316)(169 339 367 280)(170 340 368 253)(171 341 369 254)(172 342 370 255)(173 343 371 256)(174 344 372 257)(175 345 373 258)(176 346 374 259)(177 347 375 260)(178 348 376 261)(179 349 377 262)(180 350 378 263)(181 351 379 264)(182 352 380 265)(183 353 381 266)(184 354 382 267)(185 355 383 268)(186 356 384 269)(187 357 385 270)(188 358 386 271)(189 359 387 272)(190 360 388 273)(191 361 389 274)(192 362 390 275)(193 363 391 276)(194 364 392 277)(195 337 365 278)(196 338 366 279)
(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)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)

G:=sub<Sym(448)| (1,277,163,86)(2,278,164,87)(3,279,165,88)(4,280,166,89)(5,253,167,90)(6,254,168,91)(7,255,141,92)(8,256,142,93)(9,257,143,94)(10,258,144,95)(11,259,145,96)(12,260,146,97)(13,261,147,98)(14,262,148,99)(15,263,149,100)(16,264,150,101)(17,265,151,102)(18,266,152,103)(19,267,153,104)(20,268,154,105)(21,269,155,106)(22,270,156,107)(23,271,157,108)(24,272,158,109)(25,273,159,110)(26,274,160,111)(27,275,161,112)(28,276,162,85)(29,342,234,399)(30,343,235,400)(31,344,236,401)(32,345,237,402)(33,346,238,403)(34,347,239,404)(35,348,240,405)(36,349,241,406)(37,350,242,407)(38,351,243,408)(39,352,244,409)(40,353,245,410)(41,354,246,411)(42,355,247,412)(43,356,248,413)(44,357,249,414)(45,358,250,415)(46,359,251,416)(47,360,252,417)(48,361,225,418)(49,362,226,419)(50,363,227,420)(51,364,228,393)(52,337,229,394)(53,338,230,395)(54,339,231,396)(55,340,232,397)(56,341,233,398)(57,324,209,370)(58,325,210,371)(59,326,211,372)(60,327,212,373)(61,328,213,374)(62,329,214,375)(63,330,215,376)(64,331,216,377)(65,332,217,378)(66,333,218,379)(67,334,219,380)(68,335,220,381)(69,336,221,382)(70,309,222,383)(71,310,223,384)(72,311,224,385)(73,312,197,386)(74,313,198,387)(75,314,199,388)(76,315,200,389)(77,316,201,390)(78,317,202,391)(79,318,203,392)(80,319,204,365)(81,320,205,366)(82,321,206,367)(83,322,207,368)(84,323,208,369)(113,288,184,429)(114,289,185,430)(115,290,186,431)(116,291,187,432)(117,292,188,433)(118,293,189,434)(119,294,190,435)(120,295,191,436)(121,296,192,437)(122,297,193,438)(123,298,194,439)(124,299,195,440)(125,300,196,441)(126,301,169,442)(127,302,170,443)(128,303,171,444)(129,304,172,445)(130,305,173,446)(131,306,174,447)(132,307,175,448)(133,308,176,421)(134,281,177,422)(135,282,178,423)(136,283,179,424)(137,284,180,425)(138,285,181,426)(139,286,182,427)(140,287,183,428), (1,298,51,203)(2,299,52,204)(3,300,53,205)(4,301,54,206)(5,302,55,207)(6,303,56,208)(7,304,29,209)(8,305,30,210)(9,306,31,211)(10,307,32,212)(11,308,33,213)(12,281,34,214)(13,282,35,215)(14,283,36,216)(15,284,37,217)(16,285,38,218)(17,286,39,219)(18,287,40,220)(19,288,41,221)(20,289,42,222)(21,290,43,223)(22,291,44,224)(23,292,45,197)(24,293,46,198)(25,294,47,199)(26,295,48,200)(27,296,49,201)(28,297,50,202)(57,141,445,234)(58,142,446,235)(59,143,447,236)(60,144,448,237)(61,145,421,238)(62,146,422,239)(63,147,423,240)(64,148,424,241)(65,149,425,242)(66,150,426,243)(67,151,427,244)(68,152,428,245)(69,153,429,246)(70,154,430,247)(71,155,431,248)(72,156,432,249)(73,157,433,250)(74,158,434,251)(75,159,435,252)(76,160,436,225)(77,161,437,226)(78,162,438,227)(79,163,439,228)(80,164,440,229)(81,165,441,230)(82,166,442,231)(83,167,443,232)(84,168,444,233)(85,122,420,317)(86,123,393,318)(87,124,394,319)(88,125,395,320)(89,126,396,321)(90,127,397,322)(91,128,398,323)(92,129,399,324)(93,130,400,325)(94,131,401,326)(95,132,402,327)(96,133,403,328)(97,134,404,329)(98,135,405,330)(99,136,406,331)(100,137,407,332)(101,138,408,333)(102,139,409,334)(103,140,410,335)(104,113,411,336)(105,114,412,309)(106,115,413,310)(107,116,414,311)(108,117,415,312)(109,118,416,313)(110,119,417,314)(111,120,418,315)(112,121,419,316)(169,339,367,280)(170,340,368,253)(171,341,369,254)(172,342,370,255)(173,343,371,256)(174,344,372,257)(175,345,373,258)(176,346,374,259)(177,347,375,260)(178,348,376,261)(179,349,377,262)(180,350,378,263)(181,351,379,264)(182,352,380,265)(183,353,381,266)(184,354,382,267)(185,355,383,268)(186,356,384,269)(187,357,385,270)(188,358,386,271)(189,359,387,272)(190,360,388,273)(191,361,389,274)(192,362,390,275)(193,363,391,276)(194,364,392,277)(195,337,365,278)(196,338,366,279), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)>;

G:=Group( (1,277,163,86)(2,278,164,87)(3,279,165,88)(4,280,166,89)(5,253,167,90)(6,254,168,91)(7,255,141,92)(8,256,142,93)(9,257,143,94)(10,258,144,95)(11,259,145,96)(12,260,146,97)(13,261,147,98)(14,262,148,99)(15,263,149,100)(16,264,150,101)(17,265,151,102)(18,266,152,103)(19,267,153,104)(20,268,154,105)(21,269,155,106)(22,270,156,107)(23,271,157,108)(24,272,158,109)(25,273,159,110)(26,274,160,111)(27,275,161,112)(28,276,162,85)(29,342,234,399)(30,343,235,400)(31,344,236,401)(32,345,237,402)(33,346,238,403)(34,347,239,404)(35,348,240,405)(36,349,241,406)(37,350,242,407)(38,351,243,408)(39,352,244,409)(40,353,245,410)(41,354,246,411)(42,355,247,412)(43,356,248,413)(44,357,249,414)(45,358,250,415)(46,359,251,416)(47,360,252,417)(48,361,225,418)(49,362,226,419)(50,363,227,420)(51,364,228,393)(52,337,229,394)(53,338,230,395)(54,339,231,396)(55,340,232,397)(56,341,233,398)(57,324,209,370)(58,325,210,371)(59,326,211,372)(60,327,212,373)(61,328,213,374)(62,329,214,375)(63,330,215,376)(64,331,216,377)(65,332,217,378)(66,333,218,379)(67,334,219,380)(68,335,220,381)(69,336,221,382)(70,309,222,383)(71,310,223,384)(72,311,224,385)(73,312,197,386)(74,313,198,387)(75,314,199,388)(76,315,200,389)(77,316,201,390)(78,317,202,391)(79,318,203,392)(80,319,204,365)(81,320,205,366)(82,321,206,367)(83,322,207,368)(84,323,208,369)(113,288,184,429)(114,289,185,430)(115,290,186,431)(116,291,187,432)(117,292,188,433)(118,293,189,434)(119,294,190,435)(120,295,191,436)(121,296,192,437)(122,297,193,438)(123,298,194,439)(124,299,195,440)(125,300,196,441)(126,301,169,442)(127,302,170,443)(128,303,171,444)(129,304,172,445)(130,305,173,446)(131,306,174,447)(132,307,175,448)(133,308,176,421)(134,281,177,422)(135,282,178,423)(136,283,179,424)(137,284,180,425)(138,285,181,426)(139,286,182,427)(140,287,183,428), (1,298,51,203)(2,299,52,204)(3,300,53,205)(4,301,54,206)(5,302,55,207)(6,303,56,208)(7,304,29,209)(8,305,30,210)(9,306,31,211)(10,307,32,212)(11,308,33,213)(12,281,34,214)(13,282,35,215)(14,283,36,216)(15,284,37,217)(16,285,38,218)(17,286,39,219)(18,287,40,220)(19,288,41,221)(20,289,42,222)(21,290,43,223)(22,291,44,224)(23,292,45,197)(24,293,46,198)(25,294,47,199)(26,295,48,200)(27,296,49,201)(28,297,50,202)(57,141,445,234)(58,142,446,235)(59,143,447,236)(60,144,448,237)(61,145,421,238)(62,146,422,239)(63,147,423,240)(64,148,424,241)(65,149,425,242)(66,150,426,243)(67,151,427,244)(68,152,428,245)(69,153,429,246)(70,154,430,247)(71,155,431,248)(72,156,432,249)(73,157,433,250)(74,158,434,251)(75,159,435,252)(76,160,436,225)(77,161,437,226)(78,162,438,227)(79,163,439,228)(80,164,440,229)(81,165,441,230)(82,166,442,231)(83,167,443,232)(84,168,444,233)(85,122,420,317)(86,123,393,318)(87,124,394,319)(88,125,395,320)(89,126,396,321)(90,127,397,322)(91,128,398,323)(92,129,399,324)(93,130,400,325)(94,131,401,326)(95,132,402,327)(96,133,403,328)(97,134,404,329)(98,135,405,330)(99,136,406,331)(100,137,407,332)(101,138,408,333)(102,139,409,334)(103,140,410,335)(104,113,411,336)(105,114,412,309)(106,115,413,310)(107,116,414,311)(108,117,415,312)(109,118,416,313)(110,119,417,314)(111,120,418,315)(112,121,419,316)(169,339,367,280)(170,340,368,253)(171,341,369,254)(172,342,370,255)(173,343,371,256)(174,344,372,257)(175,345,373,258)(176,346,374,259)(177,347,375,260)(178,348,376,261)(179,349,377,262)(180,350,378,263)(181,351,379,264)(182,352,380,265)(183,353,381,266)(184,354,382,267)(185,355,383,268)(186,356,384,269)(187,357,385,270)(188,358,386,271)(189,359,387,272)(190,360,388,273)(191,361,389,274)(192,362,390,275)(193,363,391,276)(194,364,392,277)(195,337,365,278)(196,338,366,279), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448) );

G=PermutationGroup([(1,277,163,86),(2,278,164,87),(3,279,165,88),(4,280,166,89),(5,253,167,90),(6,254,168,91),(7,255,141,92),(8,256,142,93),(9,257,143,94),(10,258,144,95),(11,259,145,96),(12,260,146,97),(13,261,147,98),(14,262,148,99),(15,263,149,100),(16,264,150,101),(17,265,151,102),(18,266,152,103),(19,267,153,104),(20,268,154,105),(21,269,155,106),(22,270,156,107),(23,271,157,108),(24,272,158,109),(25,273,159,110),(26,274,160,111),(27,275,161,112),(28,276,162,85),(29,342,234,399),(30,343,235,400),(31,344,236,401),(32,345,237,402),(33,346,238,403),(34,347,239,404),(35,348,240,405),(36,349,241,406),(37,350,242,407),(38,351,243,408),(39,352,244,409),(40,353,245,410),(41,354,246,411),(42,355,247,412),(43,356,248,413),(44,357,249,414),(45,358,250,415),(46,359,251,416),(47,360,252,417),(48,361,225,418),(49,362,226,419),(50,363,227,420),(51,364,228,393),(52,337,229,394),(53,338,230,395),(54,339,231,396),(55,340,232,397),(56,341,233,398),(57,324,209,370),(58,325,210,371),(59,326,211,372),(60,327,212,373),(61,328,213,374),(62,329,214,375),(63,330,215,376),(64,331,216,377),(65,332,217,378),(66,333,218,379),(67,334,219,380),(68,335,220,381),(69,336,221,382),(70,309,222,383),(71,310,223,384),(72,311,224,385),(73,312,197,386),(74,313,198,387),(75,314,199,388),(76,315,200,389),(77,316,201,390),(78,317,202,391),(79,318,203,392),(80,319,204,365),(81,320,205,366),(82,321,206,367),(83,322,207,368),(84,323,208,369),(113,288,184,429),(114,289,185,430),(115,290,186,431),(116,291,187,432),(117,292,188,433),(118,293,189,434),(119,294,190,435),(120,295,191,436),(121,296,192,437),(122,297,193,438),(123,298,194,439),(124,299,195,440),(125,300,196,441),(126,301,169,442),(127,302,170,443),(128,303,171,444),(129,304,172,445),(130,305,173,446),(131,306,174,447),(132,307,175,448),(133,308,176,421),(134,281,177,422),(135,282,178,423),(136,283,179,424),(137,284,180,425),(138,285,181,426),(139,286,182,427),(140,287,183,428)], [(1,298,51,203),(2,299,52,204),(3,300,53,205),(4,301,54,206),(5,302,55,207),(6,303,56,208),(7,304,29,209),(8,305,30,210),(9,306,31,211),(10,307,32,212),(11,308,33,213),(12,281,34,214),(13,282,35,215),(14,283,36,216),(15,284,37,217),(16,285,38,218),(17,286,39,219),(18,287,40,220),(19,288,41,221),(20,289,42,222),(21,290,43,223),(22,291,44,224),(23,292,45,197),(24,293,46,198),(25,294,47,199),(26,295,48,200),(27,296,49,201),(28,297,50,202),(57,141,445,234),(58,142,446,235),(59,143,447,236),(60,144,448,237),(61,145,421,238),(62,146,422,239),(63,147,423,240),(64,148,424,241),(65,149,425,242),(66,150,426,243),(67,151,427,244),(68,152,428,245),(69,153,429,246),(70,154,430,247),(71,155,431,248),(72,156,432,249),(73,157,433,250),(74,158,434,251),(75,159,435,252),(76,160,436,225),(77,161,437,226),(78,162,438,227),(79,163,439,228),(80,164,440,229),(81,165,441,230),(82,166,442,231),(83,167,443,232),(84,168,444,233),(85,122,420,317),(86,123,393,318),(87,124,394,319),(88,125,395,320),(89,126,396,321),(90,127,397,322),(91,128,398,323),(92,129,399,324),(93,130,400,325),(94,131,401,326),(95,132,402,327),(96,133,403,328),(97,134,404,329),(98,135,405,330),(99,136,406,331),(100,137,407,332),(101,138,408,333),(102,139,409,334),(103,140,410,335),(104,113,411,336),(105,114,412,309),(106,115,413,310),(107,116,414,311),(108,117,415,312),(109,118,416,313),(110,119,417,314),(111,120,418,315),(112,121,419,316),(169,339,367,280),(170,340,368,253),(171,341,369,254),(172,342,370,255),(173,343,371,256),(174,344,372,257),(175,345,373,258),(176,346,374,259),(177,347,375,260),(178,348,376,261),(179,349,377,262),(180,350,378,263),(181,351,379,264),(182,352,380,265),(183,353,381,266),(184,354,382,267),(185,355,383,268),(186,356,384,269),(187,357,385,270),(188,358,386,271),(189,359,387,272),(190,360,388,273),(191,361,389,274),(192,362,390,275),(193,363,391,276),(194,364,392,277),(195,337,365,278),(196,338,366,279)], [(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),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)])

448 conjugacy classes

class 1 2A···2G4A···4BD7A···7F14A···14AP28A···28LX
order12···24···47···714···1428···28
size11···11···11···11···11···1

448 irreducible representations

dim111111
type++
imageC1C2C4C7C14C28
kernelC42×C28C2×C4×C28C4×C28C43C2×C42C42
# reps1756642336

Matrix representation of C42×C28 in GL3(𝔽29) generated by

2800
0280
0012
,
100
0120
0017
,
1700
020
0014
G:=sub<GL(3,GF(29))| [28,0,0,0,28,0,0,0,12],[1,0,0,0,12,0,0,0,17],[17,0,0,0,2,0,0,0,14] >;

C42×C28 in GAP, Magma, Sage, TeX

C_4^2\times C_{28}
% in TeX

G:=Group("C4^2xC28");
// GroupNames label

G:=SmallGroup(448,782);
// by ID

G=gap.SmallGroup(448,782);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,792,1192]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^28=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽