metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊4Dic7, C28.24C42, (C4×C28)⋊15C4, (C2×C42).4D7, C7⋊2(C42⋊4C4), (C4×Dic7)⋊15C4, C4.19(C4×Dic7), C14.17(C2×C42), (C22×C4).458D14, C2.3(C42⋊D7), C22.42(C4○D28), C23.265(C22×D7), C14.12(C42⋊C2), C14.C42.42C2, (C22×C14).307C23, (C22×C28).472C22, C22.17(C22×Dic7), C2.2(C23.21D14), (C22×Dic7).179C22, (C2×C4×C28).29C2, C2.6(C2×C4×Dic7), C22.51(C2×C4×D7), (C2×C4).180(C4×D7), (C2×C4×Dic7).32C2, (C2×C28).316(C2×C4), (C2×C4).60(C2×Dic7), (C2×C14).67(C4○D4), (C2×Dic7).91(C2×C4), (C2×C14).174(C22×C4), SmallGroup(448,466)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊4Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 580 in 178 conjugacy classes, 103 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C42, C22×C4, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C42, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊4C4, C4×Dic7, C4×C28, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4×Dic7, C2×C4×C28, C42⋊4Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, C4○D4, Dic7, D14, C2×C42, C42⋊C2, C4×D7, C2×Dic7, C22×D7, C42⋊4C4, C4×Dic7, C2×C4×D7, C4○D28, C22×Dic7, C42⋊D7, C2×C4×Dic7, C23.21D14, C42⋊4Dic7
►Regular action on 448 points
Generators in S448
(1 24 242 92)(2 25 243 93)(3 26 244 94)(4 27 245 95)(5 28 246 96)(6 15 247 97)(7 16 248 98)(8 17 249 85)(9 18 250 86)(10 19 251 87)(11 20 252 88)(12 21 239 89)(13 22 240 90)(14 23 241 91)(29 404 221 285)(30 405 222 286)(31 406 223 287)(32 393 224 288)(33 394 211 289)(34 395 212 290)(35 396 213 291)(36 397 214 292)(37 398 215 293)(38 399 216 294)(39 400 217 281)(40 401 218 282)(41 402 219 283)(42 403 220 284)(43 391 194 261)(44 392 195 262)(45 379 196 263)(46 380 183 264)(47 381 184 265)(48 382 185 266)(49 383 186 253)(50 384 187 254)(51 385 188 255)(52 386 189 256)(53 387 190 257)(54 388 191 258)(55 389 192 259)(56 390 193 260)(57 306 275 374)(58 307 276 375)(59 308 277 376)(60 295 278 377)(61 296 279 378)(62 297 280 365)(63 298 267 366)(64 299 268 367)(65 300 269 368)(66 301 270 369)(67 302 271 370)(68 303 272 371)(69 304 273 372)(70 305 274 373)(71 432 148 126)(72 433 149 113)(73 434 150 114)(74 421 151 115)(75 422 152 116)(76 423 153 117)(77 424 154 118)(78 425 141 119)(79 426 142 120)(80 427 143 121)(81 428 144 122)(82 429 145 123)(83 430 146 124)(84 431 147 125)(99 328 227 442)(100 329 228 443)(101 330 229 444)(102 331 230 445)(103 332 231 446)(104 333 232 447)(105 334 233 448)(106 335 234 435)(107 336 235 436)(108 323 236 437)(109 324 237 438)(110 325 238 439)(111 326 225 440)(112 327 226 441)(127 311 410 353)(128 312 411 354)(129 313 412 355)(130 314 413 356)(131 315 414 357)(132 316 415 358)(133 317 416 359)(134 318 417 360)(135 319 418 361)(136 320 419 362)(137 321 420 363)(138 322 407 364)(139 309 408 351)(140 310 409 352)(155 181 337 201)(156 182 338 202)(157 169 339 203)(158 170 340 204)(159 171 341 205)(160 172 342 206)(161 173 343 207)(162 174 344 208)(163 175 345 209)(164 176 346 210)(165 177 347 197)(166 178 348 198)(167 179 349 199)(168 180 350 200)
(1 372 30 352)(2 373 31 353)(3 374 32 354)(4 375 33 355)(5 376 34 356)(6 377 35 357)(7 378 36 358)(8 365 37 359)(9 366 38 360)(10 367 39 361)(11 368 40 362)(12 369 41 363)(13 370 42 364)(14 371 29 351)(15 60 396 131)(16 61 397 132)(17 62 398 133)(18 63 399 134)(19 64 400 135)(20 65 401 136)(21 66 402 137)(22 67 403 138)(23 68 404 139)(24 69 405 140)(25 70 406 127)(26 57 393 128)(27 58 394 129)(28 59 395 130)(43 230 210 113)(44 231 197 114)(45 232 198 115)(46 233 199 116)(47 234 200 117)(48 235 201 118)(49 236 202 119)(50 237 203 120)(51 238 204 121)(52 225 205 122)(53 226 206 123)(54 227 207 124)(55 228 208 125)(56 229 209 126)(71 390 444 163)(72 391 445 164)(73 392 446 165)(74 379 447 166)(75 380 448 167)(76 381 435 168)(77 382 436 155)(78 383 437 156)(79 384 438 157)(80 385 439 158)(81 386 440 159)(82 387 441 160)(83 388 442 161)(84 389 443 162)(85 280 293 416)(86 267 294 417)(87 268 281 418)(88 269 282 419)(89 270 283 420)(90 271 284 407)(91 272 285 408)(92 273 286 409)(93 274 287 410)(94 275 288 411)(95 276 289 412)(96 277 290 413)(97 278 291 414)(98 279 292 415)(99 173 430 191)(100 174 431 192)(101 175 432 193)(102 176 433 194)(103 177 434 195)(104 178 421 196)(105 179 422 183)(106 180 423 184)(107 181 424 185)(108 182 425 186)(109 169 426 187)(110 170 427 188)(111 171 428 189)(112 172 429 190)(141 253 323 338)(142 254 324 339)(143 255 325 340)(144 256 326 341)(145 257 327 342)(146 258 328 343)(147 259 329 344)(148 260 330 345)(149 261 331 346)(150 262 332 347)(151 263 333 348)(152 264 334 349)(153 265 335 350)(154 266 336 337)(211 313 245 307)(212 314 246 308)(213 315 247 295)(214 316 248 296)(215 317 249 297)(216 318 250 298)(217 319 251 299)(218 320 252 300)(219 321 239 301)(220 322 240 302)(221 309 241 303)(222 310 242 304)(223 311 243 305)(224 312 244 306)
(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)
(1 333 8 326)(2 332 9 325)(3 331 10 324)(4 330 11 323)(5 329 12 336)(6 328 13 335)(7 327 14 334)(15 227 22 234)(16 226 23 233)(17 225 24 232)(18 238 25 231)(19 237 26 230)(20 236 27 229)(21 235 28 228)(29 152 36 145)(30 151 37 144)(31 150 38 143)(32 149 39 142)(33 148 40 141)(34 147 41 154)(35 146 42 153)(43 418 50 411)(44 417 51 410)(45 416 52 409)(46 415 53 408)(47 414 54 407)(48 413 55 420)(49 412 56 419)(57 176 64 169)(58 175 65 182)(59 174 66 181)(60 173 67 180)(61 172 68 179)(62 171 69 178)(63 170 70 177)(71 218 78 211)(72 217 79 224)(73 216 80 223)(74 215 81 222)(75 214 82 221)(76 213 83 220)(77 212 84 219)(85 111 92 104)(86 110 93 103)(87 109 94 102)(88 108 95 101)(89 107 96 100)(90 106 97 99)(91 105 98 112)(113 400 120 393)(114 399 121 406)(115 398 122 405)(116 397 123 404)(117 396 124 403)(118 395 125 402)(119 394 126 401)(127 195 134 188)(128 194 135 187)(129 193 136 186)(130 192 137 185)(131 191 138 184)(132 190 139 183)(133 189 140 196)(155 376 162 369)(156 375 163 368)(157 374 164 367)(158 373 165 366)(159 372 166 365)(160 371 167 378)(161 370 168 377)(197 267 204 274)(198 280 205 273)(199 279 206 272)(200 278 207 271)(201 277 208 270)(202 276 209 269)(203 275 210 268)(239 436 246 443)(240 435 247 442)(241 448 248 441)(242 447 249 440)(243 446 250 439)(244 445 251 438)(245 444 252 437)(253 313 260 320)(254 312 261 319)(255 311 262 318)(256 310 263 317)(257 309 264 316)(258 322 265 315)(259 321 266 314)(281 426 288 433)(282 425 289 432)(283 424 290 431)(284 423 291 430)(285 422 292 429)(286 421 293 428)(287 434 294 427)(295 343 302 350)(296 342 303 349)(297 341 304 348)(298 340 305 347)(299 339 306 346)(300 338 307 345)(301 337 308 344)(351 380 358 387)(352 379 359 386)(353 392 360 385)(354 391 361 384)(355 390 362 383)(356 389 363 382)(357 388 364 381)
G:=sub<Sym(448)| (1,24,242,92)(2,25,243,93)(3,26,244,94)(4,27,245,95)(5,28,246,96)(6,15,247,97)(7,16,248,98)(8,17,249,85)(9,18,250,86)(10,19,251,87)(11,20,252,88)(12,21,239,89)(13,22,240,90)(14,23,241,91)(29,404,221,285)(30,405,222,286)(31,406,223,287)(32,393,224,288)(33,394,211,289)(34,395,212,290)(35,396,213,291)(36,397,214,292)(37,398,215,293)(38,399,216,294)(39,400,217,281)(40,401,218,282)(41,402,219,283)(42,403,220,284)(43,391,194,261)(44,392,195,262)(45,379,196,263)(46,380,183,264)(47,381,184,265)(48,382,185,266)(49,383,186,253)(50,384,187,254)(51,385,188,255)(52,386,189,256)(53,387,190,257)(54,388,191,258)(55,389,192,259)(56,390,193,260)(57,306,275,374)(58,307,276,375)(59,308,277,376)(60,295,278,377)(61,296,279,378)(62,297,280,365)(63,298,267,366)(64,299,268,367)(65,300,269,368)(66,301,270,369)(67,302,271,370)(68,303,272,371)(69,304,273,372)(70,305,274,373)(71,432,148,126)(72,433,149,113)(73,434,150,114)(74,421,151,115)(75,422,152,116)(76,423,153,117)(77,424,154,118)(78,425,141,119)(79,426,142,120)(80,427,143,121)(81,428,144,122)(82,429,145,123)(83,430,146,124)(84,431,147,125)(99,328,227,442)(100,329,228,443)(101,330,229,444)(102,331,230,445)(103,332,231,446)(104,333,232,447)(105,334,233,448)(106,335,234,435)(107,336,235,436)(108,323,236,437)(109,324,237,438)(110,325,238,439)(111,326,225,440)(112,327,226,441)(127,311,410,353)(128,312,411,354)(129,313,412,355)(130,314,413,356)(131,315,414,357)(132,316,415,358)(133,317,416,359)(134,318,417,360)(135,319,418,361)(136,320,419,362)(137,321,420,363)(138,322,407,364)(139,309,408,351)(140,310,409,352)(155,181,337,201)(156,182,338,202)(157,169,339,203)(158,170,340,204)(159,171,341,205)(160,172,342,206)(161,173,343,207)(162,174,344,208)(163,175,345,209)(164,176,346,210)(165,177,347,197)(166,178,348,198)(167,179,349,199)(168,180,350,200), (1,372,30,352)(2,373,31,353)(3,374,32,354)(4,375,33,355)(5,376,34,356)(6,377,35,357)(7,378,36,358)(8,365,37,359)(9,366,38,360)(10,367,39,361)(11,368,40,362)(12,369,41,363)(13,370,42,364)(14,371,29,351)(15,60,396,131)(16,61,397,132)(17,62,398,133)(18,63,399,134)(19,64,400,135)(20,65,401,136)(21,66,402,137)(22,67,403,138)(23,68,404,139)(24,69,405,140)(25,70,406,127)(26,57,393,128)(27,58,394,129)(28,59,395,130)(43,230,210,113)(44,231,197,114)(45,232,198,115)(46,233,199,116)(47,234,200,117)(48,235,201,118)(49,236,202,119)(50,237,203,120)(51,238,204,121)(52,225,205,122)(53,226,206,123)(54,227,207,124)(55,228,208,125)(56,229,209,126)(71,390,444,163)(72,391,445,164)(73,392,446,165)(74,379,447,166)(75,380,448,167)(76,381,435,168)(77,382,436,155)(78,383,437,156)(79,384,438,157)(80,385,439,158)(81,386,440,159)(82,387,441,160)(83,388,442,161)(84,389,443,162)(85,280,293,416)(86,267,294,417)(87,268,281,418)(88,269,282,419)(89,270,283,420)(90,271,284,407)(91,272,285,408)(92,273,286,409)(93,274,287,410)(94,275,288,411)(95,276,289,412)(96,277,290,413)(97,278,291,414)(98,279,292,415)(99,173,430,191)(100,174,431,192)(101,175,432,193)(102,176,433,194)(103,177,434,195)(104,178,421,196)(105,179,422,183)(106,180,423,184)(107,181,424,185)(108,182,425,186)(109,169,426,187)(110,170,427,188)(111,171,428,189)(112,172,429,190)(141,253,323,338)(142,254,324,339)(143,255,325,340)(144,256,326,341)(145,257,327,342)(146,258,328,343)(147,259,329,344)(148,260,330,345)(149,261,331,346)(150,262,332,347)(151,263,333,348)(152,264,334,349)(153,265,335,350)(154,266,336,337)(211,313,245,307)(212,314,246,308)(213,315,247,295)(214,316,248,296)(215,317,249,297)(216,318,250,298)(217,319,251,299)(218,320,252,300)(219,321,239,301)(220,322,240,302)(221,309,241,303)(222,310,242,304)(223,311,243,305)(224,312,244,306), (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), (1,333,8,326)(2,332,9,325)(3,331,10,324)(4,330,11,323)(5,329,12,336)(6,328,13,335)(7,327,14,334)(15,227,22,234)(16,226,23,233)(17,225,24,232)(18,238,25,231)(19,237,26,230)(20,236,27,229)(21,235,28,228)(29,152,36,145)(30,151,37,144)(31,150,38,143)(32,149,39,142)(33,148,40,141)(34,147,41,154)(35,146,42,153)(43,418,50,411)(44,417,51,410)(45,416,52,409)(46,415,53,408)(47,414,54,407)(48,413,55,420)(49,412,56,419)(57,176,64,169)(58,175,65,182)(59,174,66,181)(60,173,67,180)(61,172,68,179)(62,171,69,178)(63,170,70,177)(71,218,78,211)(72,217,79,224)(73,216,80,223)(74,215,81,222)(75,214,82,221)(76,213,83,220)(77,212,84,219)(85,111,92,104)(86,110,93,103)(87,109,94,102)(88,108,95,101)(89,107,96,100)(90,106,97,99)(91,105,98,112)(113,400,120,393)(114,399,121,406)(115,398,122,405)(116,397,123,404)(117,396,124,403)(118,395,125,402)(119,394,126,401)(127,195,134,188)(128,194,135,187)(129,193,136,186)(130,192,137,185)(131,191,138,184)(132,190,139,183)(133,189,140,196)(155,376,162,369)(156,375,163,368)(157,374,164,367)(158,373,165,366)(159,372,166,365)(160,371,167,378)(161,370,168,377)(197,267,204,274)(198,280,205,273)(199,279,206,272)(200,278,207,271)(201,277,208,270)(202,276,209,269)(203,275,210,268)(239,436,246,443)(240,435,247,442)(241,448,248,441)(242,447,249,440)(243,446,250,439)(244,445,251,438)(245,444,252,437)(253,313,260,320)(254,312,261,319)(255,311,262,318)(256,310,263,317)(257,309,264,316)(258,322,265,315)(259,321,266,314)(281,426,288,433)(282,425,289,432)(283,424,290,431)(284,423,291,430)(285,422,292,429)(286,421,293,428)(287,434,294,427)(295,343,302,350)(296,342,303,349)(297,341,304,348)(298,340,305,347)(299,339,306,346)(300,338,307,345)(301,337,308,344)(351,380,358,387)(352,379,359,386)(353,392,360,385)(354,391,361,384)(355,390,362,383)(356,389,363,382)(357,388,364,381)>;
G:=Group( (1,24,242,92)(2,25,243,93)(3,26,244,94)(4,27,245,95)(5,28,246,96)(6,15,247,97)(7,16,248,98)(8,17,249,85)(9,18,250,86)(10,19,251,87)(11,20,252,88)(12,21,239,89)(13,22,240,90)(14,23,241,91)(29,404,221,285)(30,405,222,286)(31,406,223,287)(32,393,224,288)(33,394,211,289)(34,395,212,290)(35,396,213,291)(36,397,214,292)(37,398,215,293)(38,399,216,294)(39,400,217,281)(40,401,218,282)(41,402,219,283)(42,403,220,284)(43,391,194,261)(44,392,195,262)(45,379,196,263)(46,380,183,264)(47,381,184,265)(48,382,185,266)(49,383,186,253)(50,384,187,254)(51,385,188,255)(52,386,189,256)(53,387,190,257)(54,388,191,258)(55,389,192,259)(56,390,193,260)(57,306,275,374)(58,307,276,375)(59,308,277,376)(60,295,278,377)(61,296,279,378)(62,297,280,365)(63,298,267,366)(64,299,268,367)(65,300,269,368)(66,301,270,369)(67,302,271,370)(68,303,272,371)(69,304,273,372)(70,305,274,373)(71,432,148,126)(72,433,149,113)(73,434,150,114)(74,421,151,115)(75,422,152,116)(76,423,153,117)(77,424,154,118)(78,425,141,119)(79,426,142,120)(80,427,143,121)(81,428,144,122)(82,429,145,123)(83,430,146,124)(84,431,147,125)(99,328,227,442)(100,329,228,443)(101,330,229,444)(102,331,230,445)(103,332,231,446)(104,333,232,447)(105,334,233,448)(106,335,234,435)(107,336,235,436)(108,323,236,437)(109,324,237,438)(110,325,238,439)(111,326,225,440)(112,327,226,441)(127,311,410,353)(128,312,411,354)(129,313,412,355)(130,314,413,356)(131,315,414,357)(132,316,415,358)(133,317,416,359)(134,318,417,360)(135,319,418,361)(136,320,419,362)(137,321,420,363)(138,322,407,364)(139,309,408,351)(140,310,409,352)(155,181,337,201)(156,182,338,202)(157,169,339,203)(158,170,340,204)(159,171,341,205)(160,172,342,206)(161,173,343,207)(162,174,344,208)(163,175,345,209)(164,176,346,210)(165,177,347,197)(166,178,348,198)(167,179,349,199)(168,180,350,200), (1,372,30,352)(2,373,31,353)(3,374,32,354)(4,375,33,355)(5,376,34,356)(6,377,35,357)(7,378,36,358)(8,365,37,359)(9,366,38,360)(10,367,39,361)(11,368,40,362)(12,369,41,363)(13,370,42,364)(14,371,29,351)(15,60,396,131)(16,61,397,132)(17,62,398,133)(18,63,399,134)(19,64,400,135)(20,65,401,136)(21,66,402,137)(22,67,403,138)(23,68,404,139)(24,69,405,140)(25,70,406,127)(26,57,393,128)(27,58,394,129)(28,59,395,130)(43,230,210,113)(44,231,197,114)(45,232,198,115)(46,233,199,116)(47,234,200,117)(48,235,201,118)(49,236,202,119)(50,237,203,120)(51,238,204,121)(52,225,205,122)(53,226,206,123)(54,227,207,124)(55,228,208,125)(56,229,209,126)(71,390,444,163)(72,391,445,164)(73,392,446,165)(74,379,447,166)(75,380,448,167)(76,381,435,168)(77,382,436,155)(78,383,437,156)(79,384,438,157)(80,385,439,158)(81,386,440,159)(82,387,441,160)(83,388,442,161)(84,389,443,162)(85,280,293,416)(86,267,294,417)(87,268,281,418)(88,269,282,419)(89,270,283,420)(90,271,284,407)(91,272,285,408)(92,273,286,409)(93,274,287,410)(94,275,288,411)(95,276,289,412)(96,277,290,413)(97,278,291,414)(98,279,292,415)(99,173,430,191)(100,174,431,192)(101,175,432,193)(102,176,433,194)(103,177,434,195)(104,178,421,196)(105,179,422,183)(106,180,423,184)(107,181,424,185)(108,182,425,186)(109,169,426,187)(110,170,427,188)(111,171,428,189)(112,172,429,190)(141,253,323,338)(142,254,324,339)(143,255,325,340)(144,256,326,341)(145,257,327,342)(146,258,328,343)(147,259,329,344)(148,260,330,345)(149,261,331,346)(150,262,332,347)(151,263,333,348)(152,264,334,349)(153,265,335,350)(154,266,336,337)(211,313,245,307)(212,314,246,308)(213,315,247,295)(214,316,248,296)(215,317,249,297)(216,318,250,298)(217,319,251,299)(218,320,252,300)(219,321,239,301)(220,322,240,302)(221,309,241,303)(222,310,242,304)(223,311,243,305)(224,312,244,306), (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), (1,333,8,326)(2,332,9,325)(3,331,10,324)(4,330,11,323)(5,329,12,336)(6,328,13,335)(7,327,14,334)(15,227,22,234)(16,226,23,233)(17,225,24,232)(18,238,25,231)(19,237,26,230)(20,236,27,229)(21,235,28,228)(29,152,36,145)(30,151,37,144)(31,150,38,143)(32,149,39,142)(33,148,40,141)(34,147,41,154)(35,146,42,153)(43,418,50,411)(44,417,51,410)(45,416,52,409)(46,415,53,408)(47,414,54,407)(48,413,55,420)(49,412,56,419)(57,176,64,169)(58,175,65,182)(59,174,66,181)(60,173,67,180)(61,172,68,179)(62,171,69,178)(63,170,70,177)(71,218,78,211)(72,217,79,224)(73,216,80,223)(74,215,81,222)(75,214,82,221)(76,213,83,220)(77,212,84,219)(85,111,92,104)(86,110,93,103)(87,109,94,102)(88,108,95,101)(89,107,96,100)(90,106,97,99)(91,105,98,112)(113,400,120,393)(114,399,121,406)(115,398,122,405)(116,397,123,404)(117,396,124,403)(118,395,125,402)(119,394,126,401)(127,195,134,188)(128,194,135,187)(129,193,136,186)(130,192,137,185)(131,191,138,184)(132,190,139,183)(133,189,140,196)(155,376,162,369)(156,375,163,368)(157,374,164,367)(158,373,165,366)(159,372,166,365)(160,371,167,378)(161,370,168,377)(197,267,204,274)(198,280,205,273)(199,279,206,272)(200,278,207,271)(201,277,208,270)(202,276,209,269)(203,275,210,268)(239,436,246,443)(240,435,247,442)(241,448,248,441)(242,447,249,440)(243,446,250,439)(244,445,251,438)(245,444,252,437)(253,313,260,320)(254,312,261,319)(255,311,262,318)(256,310,263,317)(257,309,264,316)(258,322,265,315)(259,321,266,314)(281,426,288,433)(282,425,289,432)(283,424,290,431)(284,423,291,430)(285,422,292,429)(286,421,293,428)(287,434,294,427)(295,343,302,350)(296,342,303,349)(297,341,304,348)(298,340,305,347)(299,339,306,346)(300,338,307,345)(301,337,308,344)(351,380,358,387)(352,379,359,386)(353,392,360,385)(354,391,361,384)(355,390,362,383)(356,389,363,382)(357,388,364,381) );
G=PermutationGroup([[(1,24,242,92),(2,25,243,93),(3,26,244,94),(4,27,245,95),(5,28,246,96),(6,15,247,97),(7,16,248,98),(8,17,249,85),(9,18,250,86),(10,19,251,87),(11,20,252,88),(12,21,239,89),(13,22,240,90),(14,23,241,91),(29,404,221,285),(30,405,222,286),(31,406,223,287),(32,393,224,288),(33,394,211,289),(34,395,212,290),(35,396,213,291),(36,397,214,292),(37,398,215,293),(38,399,216,294),(39,400,217,281),(40,401,218,282),(41,402,219,283),(42,403,220,284),(43,391,194,261),(44,392,195,262),(45,379,196,263),(46,380,183,264),(47,381,184,265),(48,382,185,266),(49,383,186,253),(50,384,187,254),(51,385,188,255),(52,386,189,256),(53,387,190,257),(54,388,191,258),(55,389,192,259),(56,390,193,260),(57,306,275,374),(58,307,276,375),(59,308,277,376),(60,295,278,377),(61,296,279,378),(62,297,280,365),(63,298,267,366),(64,299,268,367),(65,300,269,368),(66,301,270,369),(67,302,271,370),(68,303,272,371),(69,304,273,372),(70,305,274,373),(71,432,148,126),(72,433,149,113),(73,434,150,114),(74,421,151,115),(75,422,152,116),(76,423,153,117),(77,424,154,118),(78,425,141,119),(79,426,142,120),(80,427,143,121),(81,428,144,122),(82,429,145,123),(83,430,146,124),(84,431,147,125),(99,328,227,442),(100,329,228,443),(101,330,229,444),(102,331,230,445),(103,332,231,446),(104,333,232,447),(105,334,233,448),(106,335,234,435),(107,336,235,436),(108,323,236,437),(109,324,237,438),(110,325,238,439),(111,326,225,440),(112,327,226,441),(127,311,410,353),(128,312,411,354),(129,313,412,355),(130,314,413,356),(131,315,414,357),(132,316,415,358),(133,317,416,359),(134,318,417,360),(135,319,418,361),(136,320,419,362),(137,321,420,363),(138,322,407,364),(139,309,408,351),(140,310,409,352),(155,181,337,201),(156,182,338,202),(157,169,339,203),(158,170,340,204),(159,171,341,205),(160,172,342,206),(161,173,343,207),(162,174,344,208),(163,175,345,209),(164,176,346,210),(165,177,347,197),(166,178,348,198),(167,179,349,199),(168,180,350,200)], [(1,372,30,352),(2,373,31,353),(3,374,32,354),(4,375,33,355),(5,376,34,356),(6,377,35,357),(7,378,36,358),(8,365,37,359),(9,366,38,360),(10,367,39,361),(11,368,40,362),(12,369,41,363),(13,370,42,364),(14,371,29,351),(15,60,396,131),(16,61,397,132),(17,62,398,133),(18,63,399,134),(19,64,400,135),(20,65,401,136),(21,66,402,137),(22,67,403,138),(23,68,404,139),(24,69,405,140),(25,70,406,127),(26,57,393,128),(27,58,394,129),(28,59,395,130),(43,230,210,113),(44,231,197,114),(45,232,198,115),(46,233,199,116),(47,234,200,117),(48,235,201,118),(49,236,202,119),(50,237,203,120),(51,238,204,121),(52,225,205,122),(53,226,206,123),(54,227,207,124),(55,228,208,125),(56,229,209,126),(71,390,444,163),(72,391,445,164),(73,392,446,165),(74,379,447,166),(75,380,448,167),(76,381,435,168),(77,382,436,155),(78,383,437,156),(79,384,438,157),(80,385,439,158),(81,386,440,159),(82,387,441,160),(83,388,442,161),(84,389,443,162),(85,280,293,416),(86,267,294,417),(87,268,281,418),(88,269,282,419),(89,270,283,420),(90,271,284,407),(91,272,285,408),(92,273,286,409),(93,274,287,410),(94,275,288,411),(95,276,289,412),(96,277,290,413),(97,278,291,414),(98,279,292,415),(99,173,430,191),(100,174,431,192),(101,175,432,193),(102,176,433,194),(103,177,434,195),(104,178,421,196),(105,179,422,183),(106,180,423,184),(107,181,424,185),(108,182,425,186),(109,169,426,187),(110,170,427,188),(111,171,428,189),(112,172,429,190),(141,253,323,338),(142,254,324,339),(143,255,325,340),(144,256,326,341),(145,257,327,342),(146,258,328,343),(147,259,329,344),(148,260,330,345),(149,261,331,346),(150,262,332,347),(151,263,333,348),(152,264,334,349),(153,265,335,350),(154,266,336,337),(211,313,245,307),(212,314,246,308),(213,315,247,295),(214,316,248,296),(215,317,249,297),(216,318,250,298),(217,319,251,299),(218,320,252,300),(219,321,239,301),(220,322,240,302),(221,309,241,303),(222,310,242,304),(223,311,243,305),(224,312,244,306)], [(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)], [(1,333,8,326),(2,332,9,325),(3,331,10,324),(4,330,11,323),(5,329,12,336),(6,328,13,335),(7,327,14,334),(15,227,22,234),(16,226,23,233),(17,225,24,232),(18,238,25,231),(19,237,26,230),(20,236,27,229),(21,235,28,228),(29,152,36,145),(30,151,37,144),(31,150,38,143),(32,149,39,142),(33,148,40,141),(34,147,41,154),(35,146,42,153),(43,418,50,411),(44,417,51,410),(45,416,52,409),(46,415,53,408),(47,414,54,407),(48,413,55,420),(49,412,56,419),(57,176,64,169),(58,175,65,182),(59,174,66,181),(60,173,67,180),(61,172,68,179),(62,171,69,178),(63,170,70,177),(71,218,78,211),(72,217,79,224),(73,216,80,223),(74,215,81,222),(75,214,82,221),(76,213,83,220),(77,212,84,219),(85,111,92,104),(86,110,93,103),(87,109,94,102),(88,108,95,101),(89,107,96,100),(90,106,97,99),(91,105,98,112),(113,400,120,393),(114,399,121,406),(115,398,122,405),(116,397,123,404),(117,396,124,403),(118,395,125,402),(119,394,126,401),(127,195,134,188),(128,194,135,187),(129,193,136,186),(130,192,137,185),(131,191,138,184),(132,190,139,183),(133,189,140,196),(155,376,162,369),(156,375,163,368),(157,374,164,367),(158,373,165,366),(159,372,166,365),(160,371,167,378),(161,370,168,377),(197,267,204,274),(198,280,205,273),(199,279,206,272),(200,278,207,271),(201,277,208,270),(202,276,209,269),(203,275,210,268),(239,436,246,443),(240,435,247,442),(241,448,248,441),(242,447,249,440),(243,446,250,439),(244,445,251,438),(245,444,252,437),(253,313,260,320),(254,312,261,319),(255,311,262,318),(256,310,263,317),(257,309,264,316),(258,322,265,315),(259,321,266,314),(281,426,288,433),(282,425,289,432),(283,424,290,431),(284,423,291,430),(285,422,292,429),(286,421,293,428),(287,434,294,427),(295,343,302,350),(296,342,303,349),(297,341,304,348),(298,340,305,347),(299,339,306,346),(300,338,307,345),(301,337,308,344),(351,380,358,387),(352,379,359,386),(353,392,360,385),(354,391,361,384),(355,390,362,383),(356,389,363,382),(357,388,364,381)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D7 | C4○D4 | Dic7 | D14 | C4×D7 | C4○D28 |
| kernel | C42⋊4Dic7 | C14.C42 | C2×C4×Dic7 | C2×C4×C28 | C4×Dic7 | C4×C28 | C2×C42 | C2×C14 | C42 | C22×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 16 | 8 | 3 | 8 | 12 | 9 | 24 | 48 |
Matrix representation of C42⋊4Dic7 ►in GL6(𝔽29)
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 1 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 0 | 0 | 0 | 20 | 8 |
| 21 | 3 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 27 | 0 | 0 |
| 0 | 0 | 27 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [17,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,1,0,0,0,0,27,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,20,0,0,0,0,1,8],[21,8,0,0,0,0,3,8,0,0,0,0,0,0,16,27,0,0,0,0,27,13,0,0,0,0,0,0,28,0,0,0,0,0,1,1] >;
C42⋊4Dic7 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4{\rm Dic}_7 % in TeX
G:=Group("C4^2:4Dic7"); // GroupNames label
G:=SmallGroup(448,466);
// by ID
G=gap.SmallGroup(448,466);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,758,100,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations