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G = C14×C42.C2  order 448 = 26·7

Direct product of C14 and C42.C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C14×C42.C2, C4.9(Q8×C14), C28.98(C2×Q8), (C2×C28).80Q8, (C2×C42).18C14, C42.88(C2×C14), C22.18(Q8×C14), C14.58(C22×Q8), (C4×C28).372C22, (C2×C28).660C23, (C2×C14).347C24, C22.21(C23×C14), C23.71(C22×C14), (C22×C14).469C23, (C22×C28).509C22, C2.4(Q8×C2×C14), (C2×C4×C28).41C2, (C2×C4⋊C4).18C14, (C14×C4⋊C4).47C2, (C2×C4).22(C7×Q8), C4⋊C4.63(C2×C14), C2.10(C14×C4○D4), C14.229(C2×C4○D4), (C2×C14).116(C2×Q8), C22.33(C7×C4○D4), (C7×C4⋊C4).386C22, (C2×C4).15(C22×C14), (C2×C14).233(C4○D4), (C22×C4).101(C2×C14), SmallGroup(448,1310)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C14×C42.C2
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C7×C42.C2 — C14×C42.C2
Lower central
C1 — C22 — C14×C42.C2
Upper central
C1 — C22×C14 — C14×C42.C2

Generators and relations for C14×C42.C2
 G = < a,b,c,d | a14=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 274 in 226 conjugacy classes, 178 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C42.C2, C2×C28, C2×C28, C22×C14, C2×C42.C2, C4×C28, C7×C4⋊C4, C22×C28, C22×C28, C2×C4×C28, C14×C4⋊C4, C7×C42.C2, C14×C42.C2
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C4○D4, C24, C2×C14, C42.C2, C22×Q8, C2×C4○D4, C7×Q8, C22×C14, C2×C42.C2, Q8×C14, C7×C4○D4, C23×C14, C7×C42.C2, Q8×C2×C14, C14×C4○D4, C14×C42.C2

Smallest permutation representation of C14×C42.C2
►Regular action on 448 points
Generators in S448
(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)

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

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

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

196 conjugacy classes

class 1 2A···2G4A···4L4M···4T7A···7F14A···14AP28A···28BT28BU···28DP
order12···24···44···47···714···1428···2828···28
size11···12···24···41···11···12···24···4

196 irreducible representations

dim111111112222
type++++-
imageC1C2C2C2C7C14C14C14Q8C4○D4C7×Q8C7×C4○D4
kernelC14×C42.C2C2×C4×C28C14×C4⋊C4C7×C42.C2C2×C42.C2C2×C42C2×C4⋊C4C42.C2C2×C28C2×C14C2×C4C22
# reps1168663648482448

Matrix representation of C14×C42.C2 ►in GL5(𝔽29)

280000
022000
002200
00010
00001
,
280000
00100
01000
000120
000012
,
10000
012000
001200
000172
000112
,
280000
013500
0241600
0002122
00098

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,17,1,0,0,0,2,12],[28,0,0,0,0,0,13,24,0,0,0,5,16,0,0,0,0,0,21,9,0,0,0,22,8] >;
 

C14×C42.C2 in GAP, Magma, Sage, TeX 

C_{14}\times C_4^2.C_2
 
% in TeX
 
G:=Group("C14xC4^2.C2");
 
// GroupNames label
 
G:=SmallGroup(448,1310);
 
// by ID
 
G=gap.SmallGroup(448,1310);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,1576,4790,604]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^14=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
 
// generators/relations
 

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×
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ℤ
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