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## G = C42×C28order 448 = 26·7

### Abelian group of type [4,4,28]

Aliases: C42×C28, SmallGroup(448,782)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42×C28
 Chief series C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C2×C4×C28 — C42×C28
 Lower central C1 — C42×C28
 Upper central 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, C4, C22, C7, C2×C4, C23, C14, C42, C22×C4, C28, C2×C14, C2×C42, C2×C28, C22×C14, C43, C4×C28, C22×C28, C2×C4×C28, C42×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, C22×C4, C28, C2×C14, C2×C42, C2×C28, C22×C14, C43, C4×C28, C22×C28, C2×C4×C28, C42×C28

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

448 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C4 C7 C14 C28 kernel C42×C28 C2×C4×C28 C4×C28 C43 C2×C42 C42 # reps 1 7 56 6 42 336

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

 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
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

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