metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊8Dic7, C28⋊5(C4⋊C4), (C4×C28)⋊10C4, C4⋊2(C4⋊Dic7), (C2×C4).90D28, C14.8(C4⋊Q8), (C2×C28).53Q8, C7⋊1(C42⋊9C4), (C2×C28).383D4, (C2×C42).16D7, C2.1(C28⋊4D4), C14.9(C4⋊1D4), C2.3(C28⋊2Q8), (C2×C4).45Dic14, C22.36(C2×D28), (C22×C4).419D14, C22.21(C2×Dic14), C23.268(C22×D7), (C22×C14).310C23, (C22×C28).513C22, C22.36(C22×Dic7), (C22×Dic7).29C22, (C2×C4×C28).11C2, C14.27(C2×C4⋊C4), C2.5(C2×C4⋊Dic7), (C2×C14).28(C2×Q8), (C2×C28).296(C2×C4), (C2×C14).146(C2×D4), (C2×C4⋊Dic7).15C2, (C2×C4).79(C2×Dic7), (C2×C14).175(C22×C4), SmallGroup(448,469)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊8Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 708 in 186 conjugacy classes, 119 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C2×Dic7, C2×C28, C22×C14, C42⋊9C4, C4⋊Dic7, C4×C28, C22×Dic7, C22×C28, C2×C4⋊Dic7, C2×C4×C28, C42⋊8Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic7, D14, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, Dic14, D28, C2×Dic7, C22×D7, C42⋊9C4, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C28⋊2Q8, C28⋊4D4, C2×C4⋊Dic7, C42⋊8Dic7
►Regular action on 448 points
Generators in S448
(1 132 402 172)(2 133 403 173)(3 134 404 174)(4 135 405 175)(5 136 406 176)(6 137 393 177)(7 138 394 178)(8 139 395 179)(9 140 396 180)(10 127 397 181)(11 128 398 182)(12 129 399 169)(13 130 400 170)(14 131 401 171)(15 235 199 320)(16 236 200 321)(17 237 201 322)(18 238 202 309)(19 225 203 310)(20 226 204 311)(21 227 205 312)(22 228 206 313)(23 229 207 314)(24 230 208 315)(25 231 209 316)(26 232 210 317)(27 233 197 318)(28 234 198 319)(29 351 373 390)(30 352 374 391)(31 353 375 392)(32 354 376 379)(33 355 377 380)(34 356 378 381)(35 357 365 382)(36 358 366 383)(37 359 367 384)(38 360 368 385)(39 361 369 386)(40 362 370 387)(41 363 371 388)(42 364 372 389)(43 107 184 336)(44 108 185 323)(45 109 186 324)(46 110 187 325)(47 111 188 326)(48 112 189 327)(49 99 190 328)(50 100 191 329)(51 101 192 330)(52 102 193 331)(53 103 194 332)(54 104 195 333)(55 105 196 334)(56 106 183 335)(57 243 432 280)(58 244 433 267)(59 245 434 268)(60 246 421 269)(61 247 422 270)(62 248 423 271)(63 249 424 272)(64 250 425 273)(65 251 426 274)(66 252 427 275)(67 239 428 276)(68 240 429 277)(69 241 430 278)(70 242 431 279)(71 445 164 223)(72 446 165 224)(73 447 166 211)(74 448 167 212)(75 435 168 213)(76 436 155 214)(77 437 156 215)(78 438 157 216)(79 439 158 217)(80 440 159 218)(81 441 160 219)(82 442 161 220)(83 443 162 221)(84 444 163 222)(85 286 254 416)(86 287 255 417)(87 288 256 418)(88 289 257 419)(89 290 258 420)(90 291 259 407)(91 292 260 408)(92 293 261 409)(93 294 262 410)(94 281 263 411)(95 282 264 412)(96 283 265 413)(97 284 266 414)(98 285 253 415)(113 337 303 146)(114 338 304 147)(115 339 305 148)(116 340 306 149)(117 341 307 150)(118 342 308 151)(119 343 295 152)(120 344 296 153)(121 345 297 154)(122 346 298 141)(123 347 299 142)(124 348 300 143)(125 349 301 144)(126 350 302 145)
(1 183 351 163)(2 184 352 164)(3 185 353 165)(4 186 354 166)(5 187 355 167)(6 188 356 168)(7 189 357 155)(8 190 358 156)(9 191 359 157)(10 192 360 158)(11 193 361 159)(12 194 362 160)(13 195 363 161)(14 196 364 162)(15 142 251 255)(16 143 252 256)(17 144 239 257)(18 145 240 258)(19 146 241 259)(20 147 242 260)(21 148 243 261)(22 149 244 262)(23 150 245 263)(24 151 246 264)(25 152 247 265)(26 153 248 266)(27 154 249 253)(28 141 250 254)(29 444 172 106)(30 445 173 107)(31 446 174 108)(32 447 175 109)(33 448 176 110)(34 435 177 111)(35 436 178 112)(36 437 179 99)(37 438 180 100)(38 439 181 101)(39 440 182 102)(40 441 169 103)(41 442 170 104)(42 443 171 105)(43 391 71 403)(44 392 72 404)(45 379 73 405)(46 380 74 406)(47 381 75 393)(48 382 76 394)(49 383 77 395)(50 384 78 396)(51 385 79 397)(52 386 80 398)(53 387 81 399)(54 388 82 400)(55 389 83 401)(56 390 84 402)(57 293 312 305)(58 294 313 306)(59 281 314 307)(60 282 315 308)(61 283 316 295)(62 284 317 296)(63 285 318 297)(64 286 319 298)(65 287 320 299)(66 288 321 300)(67 289 322 301)(68 290 309 302)(69 291 310 303)(70 292 311 304)(85 198 346 273)(86 199 347 274)(87 200 348 275)(88 201 349 276)(89 202 350 277)(90 203 337 278)(91 204 338 279)(92 205 339 280)(93 206 340 267)(94 207 341 268)(95 208 342 269)(96 209 343 270)(97 210 344 271)(98 197 345 272)(113 430 407 225)(114 431 408 226)(115 432 409 227)(116 433 410 228)(117 434 411 229)(118 421 412 230)(119 422 413 231)(120 423 414 232)(121 424 415 233)(122 425 416 234)(123 426 417 235)(124 427 418 236)(125 428 419 237)(126 429 420 238)(127 330 368 217)(128 331 369 218)(129 332 370 219)(130 333 371 220)(131 334 372 221)(132 335 373 222)(133 336 374 223)(134 323 375 224)(135 324 376 211)(136 325 377 212)(137 326 378 213)(138 327 365 214)(139 328 366 215)(140 329 367 216)
(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)
(1 208 8 201)(2 207 9 200)(3 206 10 199)(4 205 11 198)(5 204 12 197)(6 203 13 210)(7 202 14 209)(15 404 22 397)(16 403 23 396)(17 402 24 395)(18 401 25 394)(19 400 26 393)(20 399 27 406)(21 398 28 405)(29 60 36 67)(30 59 37 66)(31 58 38 65)(32 57 39 64)(33 70 40 63)(34 69 41 62)(35 68 42 61)(43 263 50 256)(44 262 51 255)(45 261 52 254)(46 260 53 253)(47 259 54 266)(48 258 55 265)(49 257 56 264)(71 150 78 143)(72 149 79 142)(73 148 80 141)(74 147 81 154)(75 146 82 153)(76 145 83 152)(77 144 84 151)(85 186 92 193)(86 185 93 192)(87 184 94 191)(88 183 95 190)(89 196 96 189)(90 195 97 188)(91 194 98 187)(99 289 106 282)(100 288 107 281)(101 287 108 294)(102 286 109 293)(103 285 110 292)(104 284 111 291)(105 283 112 290)(113 220 120 213)(114 219 121 212)(115 218 122 211)(116 217 123 224)(117 216 124 223)(118 215 125 222)(119 214 126 221)(127 235 134 228)(128 234 135 227)(129 233 136 226)(130 232 137 225)(131 231 138 238)(132 230 139 237)(133 229 140 236)(155 350 162 343)(156 349 163 342)(157 348 164 341)(158 347 165 340)(159 346 166 339)(160 345 167 338)(161 344 168 337)(169 318 176 311)(170 317 177 310)(171 316 178 309)(172 315 179 322)(173 314 180 321)(174 313 181 320)(175 312 182 319)(239 390 246 383)(240 389 247 382)(241 388 248 381)(242 387 249 380)(243 386 250 379)(244 385 251 392)(245 384 252 391)(267 360 274 353)(268 359 275 352)(269 358 276 351)(270 357 277 364)(271 356 278 363)(272 355 279 362)(273 354 280 361)(295 436 302 443)(296 435 303 442)(297 448 304 441)(298 447 305 440)(299 446 306 439)(300 445 307 438)(301 444 308 437)(323 410 330 417)(324 409 331 416)(325 408 332 415)(326 407 333 414)(327 420 334 413)(328 419 335 412)(329 418 336 411)(365 429 372 422)(366 428 373 421)(367 427 374 434)(368 426 375 433)(369 425 376 432)(370 424 377 431)(371 423 378 430)
G:=sub<Sym(448)| (1,132,402,172)(2,133,403,173)(3,134,404,174)(4,135,405,175)(5,136,406,176)(6,137,393,177)(7,138,394,178)(8,139,395,179)(9,140,396,180)(10,127,397,181)(11,128,398,182)(12,129,399,169)(13,130,400,170)(14,131,401,171)(15,235,199,320)(16,236,200,321)(17,237,201,322)(18,238,202,309)(19,225,203,310)(20,226,204,311)(21,227,205,312)(22,228,206,313)(23,229,207,314)(24,230,208,315)(25,231,209,316)(26,232,210,317)(27,233,197,318)(28,234,198,319)(29,351,373,390)(30,352,374,391)(31,353,375,392)(32,354,376,379)(33,355,377,380)(34,356,378,381)(35,357,365,382)(36,358,366,383)(37,359,367,384)(38,360,368,385)(39,361,369,386)(40,362,370,387)(41,363,371,388)(42,364,372,389)(43,107,184,336)(44,108,185,323)(45,109,186,324)(46,110,187,325)(47,111,188,326)(48,112,189,327)(49,99,190,328)(50,100,191,329)(51,101,192,330)(52,102,193,331)(53,103,194,332)(54,104,195,333)(55,105,196,334)(56,106,183,335)(57,243,432,280)(58,244,433,267)(59,245,434,268)(60,246,421,269)(61,247,422,270)(62,248,423,271)(63,249,424,272)(64,250,425,273)(65,251,426,274)(66,252,427,275)(67,239,428,276)(68,240,429,277)(69,241,430,278)(70,242,431,279)(71,445,164,223)(72,446,165,224)(73,447,166,211)(74,448,167,212)(75,435,168,213)(76,436,155,214)(77,437,156,215)(78,438,157,216)(79,439,158,217)(80,440,159,218)(81,441,160,219)(82,442,161,220)(83,443,162,221)(84,444,163,222)(85,286,254,416)(86,287,255,417)(87,288,256,418)(88,289,257,419)(89,290,258,420)(90,291,259,407)(91,292,260,408)(92,293,261,409)(93,294,262,410)(94,281,263,411)(95,282,264,412)(96,283,265,413)(97,284,266,414)(98,285,253,415)(113,337,303,146)(114,338,304,147)(115,339,305,148)(116,340,306,149)(117,341,307,150)(118,342,308,151)(119,343,295,152)(120,344,296,153)(121,345,297,154)(122,346,298,141)(123,347,299,142)(124,348,300,143)(125,349,301,144)(126,350,302,145), (1,183,351,163)(2,184,352,164)(3,185,353,165)(4,186,354,166)(5,187,355,167)(6,188,356,168)(7,189,357,155)(8,190,358,156)(9,191,359,157)(10,192,360,158)(11,193,361,159)(12,194,362,160)(13,195,363,161)(14,196,364,162)(15,142,251,255)(16,143,252,256)(17,144,239,257)(18,145,240,258)(19,146,241,259)(20,147,242,260)(21,148,243,261)(22,149,244,262)(23,150,245,263)(24,151,246,264)(25,152,247,265)(26,153,248,266)(27,154,249,253)(28,141,250,254)(29,444,172,106)(30,445,173,107)(31,446,174,108)(32,447,175,109)(33,448,176,110)(34,435,177,111)(35,436,178,112)(36,437,179,99)(37,438,180,100)(38,439,181,101)(39,440,182,102)(40,441,169,103)(41,442,170,104)(42,443,171,105)(43,391,71,403)(44,392,72,404)(45,379,73,405)(46,380,74,406)(47,381,75,393)(48,382,76,394)(49,383,77,395)(50,384,78,396)(51,385,79,397)(52,386,80,398)(53,387,81,399)(54,388,82,400)(55,389,83,401)(56,390,84,402)(57,293,312,305)(58,294,313,306)(59,281,314,307)(60,282,315,308)(61,283,316,295)(62,284,317,296)(63,285,318,297)(64,286,319,298)(65,287,320,299)(66,288,321,300)(67,289,322,301)(68,290,309,302)(69,291,310,303)(70,292,311,304)(85,198,346,273)(86,199,347,274)(87,200,348,275)(88,201,349,276)(89,202,350,277)(90,203,337,278)(91,204,338,279)(92,205,339,280)(93,206,340,267)(94,207,341,268)(95,208,342,269)(96,209,343,270)(97,210,344,271)(98,197,345,272)(113,430,407,225)(114,431,408,226)(115,432,409,227)(116,433,410,228)(117,434,411,229)(118,421,412,230)(119,422,413,231)(120,423,414,232)(121,424,415,233)(122,425,416,234)(123,426,417,235)(124,427,418,236)(125,428,419,237)(126,429,420,238)(127,330,368,217)(128,331,369,218)(129,332,370,219)(130,333,371,220)(131,334,372,221)(132,335,373,222)(133,336,374,223)(134,323,375,224)(135,324,376,211)(136,325,377,212)(137,326,378,213)(138,327,365,214)(139,328,366,215)(140,329,367,216), (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), (1,208,8,201)(2,207,9,200)(3,206,10,199)(4,205,11,198)(5,204,12,197)(6,203,13,210)(7,202,14,209)(15,404,22,397)(16,403,23,396)(17,402,24,395)(18,401,25,394)(19,400,26,393)(20,399,27,406)(21,398,28,405)(29,60,36,67)(30,59,37,66)(31,58,38,65)(32,57,39,64)(33,70,40,63)(34,69,41,62)(35,68,42,61)(43,263,50,256)(44,262,51,255)(45,261,52,254)(46,260,53,253)(47,259,54,266)(48,258,55,265)(49,257,56,264)(71,150,78,143)(72,149,79,142)(73,148,80,141)(74,147,81,154)(75,146,82,153)(76,145,83,152)(77,144,84,151)(85,186,92,193)(86,185,93,192)(87,184,94,191)(88,183,95,190)(89,196,96,189)(90,195,97,188)(91,194,98,187)(99,289,106,282)(100,288,107,281)(101,287,108,294)(102,286,109,293)(103,285,110,292)(104,284,111,291)(105,283,112,290)(113,220,120,213)(114,219,121,212)(115,218,122,211)(116,217,123,224)(117,216,124,223)(118,215,125,222)(119,214,126,221)(127,235,134,228)(128,234,135,227)(129,233,136,226)(130,232,137,225)(131,231,138,238)(132,230,139,237)(133,229,140,236)(155,350,162,343)(156,349,163,342)(157,348,164,341)(158,347,165,340)(159,346,166,339)(160,345,167,338)(161,344,168,337)(169,318,176,311)(170,317,177,310)(171,316,178,309)(172,315,179,322)(173,314,180,321)(174,313,181,320)(175,312,182,319)(239,390,246,383)(240,389,247,382)(241,388,248,381)(242,387,249,380)(243,386,250,379)(244,385,251,392)(245,384,252,391)(267,360,274,353)(268,359,275,352)(269,358,276,351)(270,357,277,364)(271,356,278,363)(272,355,279,362)(273,354,280,361)(295,436,302,443)(296,435,303,442)(297,448,304,441)(298,447,305,440)(299,446,306,439)(300,445,307,438)(301,444,308,437)(323,410,330,417)(324,409,331,416)(325,408,332,415)(326,407,333,414)(327,420,334,413)(328,419,335,412)(329,418,336,411)(365,429,372,422)(366,428,373,421)(367,427,374,434)(368,426,375,433)(369,425,376,432)(370,424,377,431)(371,423,378,430)>;
G:=Group( (1,132,402,172)(2,133,403,173)(3,134,404,174)(4,135,405,175)(5,136,406,176)(6,137,393,177)(7,138,394,178)(8,139,395,179)(9,140,396,180)(10,127,397,181)(11,128,398,182)(12,129,399,169)(13,130,400,170)(14,131,401,171)(15,235,199,320)(16,236,200,321)(17,237,201,322)(18,238,202,309)(19,225,203,310)(20,226,204,311)(21,227,205,312)(22,228,206,313)(23,229,207,314)(24,230,208,315)(25,231,209,316)(26,232,210,317)(27,233,197,318)(28,234,198,319)(29,351,373,390)(30,352,374,391)(31,353,375,392)(32,354,376,379)(33,355,377,380)(34,356,378,381)(35,357,365,382)(36,358,366,383)(37,359,367,384)(38,360,368,385)(39,361,369,386)(40,362,370,387)(41,363,371,388)(42,364,372,389)(43,107,184,336)(44,108,185,323)(45,109,186,324)(46,110,187,325)(47,111,188,326)(48,112,189,327)(49,99,190,328)(50,100,191,329)(51,101,192,330)(52,102,193,331)(53,103,194,332)(54,104,195,333)(55,105,196,334)(56,106,183,335)(57,243,432,280)(58,244,433,267)(59,245,434,268)(60,246,421,269)(61,247,422,270)(62,248,423,271)(63,249,424,272)(64,250,425,273)(65,251,426,274)(66,252,427,275)(67,239,428,276)(68,240,429,277)(69,241,430,278)(70,242,431,279)(71,445,164,223)(72,446,165,224)(73,447,166,211)(74,448,167,212)(75,435,168,213)(76,436,155,214)(77,437,156,215)(78,438,157,216)(79,439,158,217)(80,440,159,218)(81,441,160,219)(82,442,161,220)(83,443,162,221)(84,444,163,222)(85,286,254,416)(86,287,255,417)(87,288,256,418)(88,289,257,419)(89,290,258,420)(90,291,259,407)(91,292,260,408)(92,293,261,409)(93,294,262,410)(94,281,263,411)(95,282,264,412)(96,283,265,413)(97,284,266,414)(98,285,253,415)(113,337,303,146)(114,338,304,147)(115,339,305,148)(116,340,306,149)(117,341,307,150)(118,342,308,151)(119,343,295,152)(120,344,296,153)(121,345,297,154)(122,346,298,141)(123,347,299,142)(124,348,300,143)(125,349,301,144)(126,350,302,145), (1,183,351,163)(2,184,352,164)(3,185,353,165)(4,186,354,166)(5,187,355,167)(6,188,356,168)(7,189,357,155)(8,190,358,156)(9,191,359,157)(10,192,360,158)(11,193,361,159)(12,194,362,160)(13,195,363,161)(14,196,364,162)(15,142,251,255)(16,143,252,256)(17,144,239,257)(18,145,240,258)(19,146,241,259)(20,147,242,260)(21,148,243,261)(22,149,244,262)(23,150,245,263)(24,151,246,264)(25,152,247,265)(26,153,248,266)(27,154,249,253)(28,141,250,254)(29,444,172,106)(30,445,173,107)(31,446,174,108)(32,447,175,109)(33,448,176,110)(34,435,177,111)(35,436,178,112)(36,437,179,99)(37,438,180,100)(38,439,181,101)(39,440,182,102)(40,441,169,103)(41,442,170,104)(42,443,171,105)(43,391,71,403)(44,392,72,404)(45,379,73,405)(46,380,74,406)(47,381,75,393)(48,382,76,394)(49,383,77,395)(50,384,78,396)(51,385,79,397)(52,386,80,398)(53,387,81,399)(54,388,82,400)(55,389,83,401)(56,390,84,402)(57,293,312,305)(58,294,313,306)(59,281,314,307)(60,282,315,308)(61,283,316,295)(62,284,317,296)(63,285,318,297)(64,286,319,298)(65,287,320,299)(66,288,321,300)(67,289,322,301)(68,290,309,302)(69,291,310,303)(70,292,311,304)(85,198,346,273)(86,199,347,274)(87,200,348,275)(88,201,349,276)(89,202,350,277)(90,203,337,278)(91,204,338,279)(92,205,339,280)(93,206,340,267)(94,207,341,268)(95,208,342,269)(96,209,343,270)(97,210,344,271)(98,197,345,272)(113,430,407,225)(114,431,408,226)(115,432,409,227)(116,433,410,228)(117,434,411,229)(118,421,412,230)(119,422,413,231)(120,423,414,232)(121,424,415,233)(122,425,416,234)(123,426,417,235)(124,427,418,236)(125,428,419,237)(126,429,420,238)(127,330,368,217)(128,331,369,218)(129,332,370,219)(130,333,371,220)(131,334,372,221)(132,335,373,222)(133,336,374,223)(134,323,375,224)(135,324,376,211)(136,325,377,212)(137,326,378,213)(138,327,365,214)(139,328,366,215)(140,329,367,216), (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), (1,208,8,201)(2,207,9,200)(3,206,10,199)(4,205,11,198)(5,204,12,197)(6,203,13,210)(7,202,14,209)(15,404,22,397)(16,403,23,396)(17,402,24,395)(18,401,25,394)(19,400,26,393)(20,399,27,406)(21,398,28,405)(29,60,36,67)(30,59,37,66)(31,58,38,65)(32,57,39,64)(33,70,40,63)(34,69,41,62)(35,68,42,61)(43,263,50,256)(44,262,51,255)(45,261,52,254)(46,260,53,253)(47,259,54,266)(48,258,55,265)(49,257,56,264)(71,150,78,143)(72,149,79,142)(73,148,80,141)(74,147,81,154)(75,146,82,153)(76,145,83,152)(77,144,84,151)(85,186,92,193)(86,185,93,192)(87,184,94,191)(88,183,95,190)(89,196,96,189)(90,195,97,188)(91,194,98,187)(99,289,106,282)(100,288,107,281)(101,287,108,294)(102,286,109,293)(103,285,110,292)(104,284,111,291)(105,283,112,290)(113,220,120,213)(114,219,121,212)(115,218,122,211)(116,217,123,224)(117,216,124,223)(118,215,125,222)(119,214,126,221)(127,235,134,228)(128,234,135,227)(129,233,136,226)(130,232,137,225)(131,231,138,238)(132,230,139,237)(133,229,140,236)(155,350,162,343)(156,349,163,342)(157,348,164,341)(158,347,165,340)(159,346,166,339)(160,345,167,338)(161,344,168,337)(169,318,176,311)(170,317,177,310)(171,316,178,309)(172,315,179,322)(173,314,180,321)(174,313,181,320)(175,312,182,319)(239,390,246,383)(240,389,247,382)(241,388,248,381)(242,387,249,380)(243,386,250,379)(244,385,251,392)(245,384,252,391)(267,360,274,353)(268,359,275,352)(269,358,276,351)(270,357,277,364)(271,356,278,363)(272,355,279,362)(273,354,280,361)(295,436,302,443)(296,435,303,442)(297,448,304,441)(298,447,305,440)(299,446,306,439)(300,445,307,438)(301,444,308,437)(323,410,330,417)(324,409,331,416)(325,408,332,415)(326,407,333,414)(327,420,334,413)(328,419,335,412)(329,418,336,411)(365,429,372,422)(366,428,373,421)(367,427,374,434)(368,426,375,433)(369,425,376,432)(370,424,377,431)(371,423,378,430) );
G=PermutationGroup([[(1,132,402,172),(2,133,403,173),(3,134,404,174),(4,135,405,175),(5,136,406,176),(6,137,393,177),(7,138,394,178),(8,139,395,179),(9,140,396,180),(10,127,397,181),(11,128,398,182),(12,129,399,169),(13,130,400,170),(14,131,401,171),(15,235,199,320),(16,236,200,321),(17,237,201,322),(18,238,202,309),(19,225,203,310),(20,226,204,311),(21,227,205,312),(22,228,206,313),(23,229,207,314),(24,230,208,315),(25,231,209,316),(26,232,210,317),(27,233,197,318),(28,234,198,319),(29,351,373,390),(30,352,374,391),(31,353,375,392),(32,354,376,379),(33,355,377,380),(34,356,378,381),(35,357,365,382),(36,358,366,383),(37,359,367,384),(38,360,368,385),(39,361,369,386),(40,362,370,387),(41,363,371,388),(42,364,372,389),(43,107,184,336),(44,108,185,323),(45,109,186,324),(46,110,187,325),(47,111,188,326),(48,112,189,327),(49,99,190,328),(50,100,191,329),(51,101,192,330),(52,102,193,331),(53,103,194,332),(54,104,195,333),(55,105,196,334),(56,106,183,335),(57,243,432,280),(58,244,433,267),(59,245,434,268),(60,246,421,269),(61,247,422,270),(62,248,423,271),(63,249,424,272),(64,250,425,273),(65,251,426,274),(66,252,427,275),(67,239,428,276),(68,240,429,277),(69,241,430,278),(70,242,431,279),(71,445,164,223),(72,446,165,224),(73,447,166,211),(74,448,167,212),(75,435,168,213),(76,436,155,214),(77,437,156,215),(78,438,157,216),(79,439,158,217),(80,440,159,218),(81,441,160,219),(82,442,161,220),(83,443,162,221),(84,444,163,222),(85,286,254,416),(86,287,255,417),(87,288,256,418),(88,289,257,419),(89,290,258,420),(90,291,259,407),(91,292,260,408),(92,293,261,409),(93,294,262,410),(94,281,263,411),(95,282,264,412),(96,283,265,413),(97,284,266,414),(98,285,253,415),(113,337,303,146),(114,338,304,147),(115,339,305,148),(116,340,306,149),(117,341,307,150),(118,342,308,151),(119,343,295,152),(120,344,296,153),(121,345,297,154),(122,346,298,141),(123,347,299,142),(124,348,300,143),(125,349,301,144),(126,350,302,145)], [(1,183,351,163),(2,184,352,164),(3,185,353,165),(4,186,354,166),(5,187,355,167),(6,188,356,168),(7,189,357,155),(8,190,358,156),(9,191,359,157),(10,192,360,158),(11,193,361,159),(12,194,362,160),(13,195,363,161),(14,196,364,162),(15,142,251,255),(16,143,252,256),(17,144,239,257),(18,145,240,258),(19,146,241,259),(20,147,242,260),(21,148,243,261),(22,149,244,262),(23,150,245,263),(24,151,246,264),(25,152,247,265),(26,153,248,266),(27,154,249,253),(28,141,250,254),(29,444,172,106),(30,445,173,107),(31,446,174,108),(32,447,175,109),(33,448,176,110),(34,435,177,111),(35,436,178,112),(36,437,179,99),(37,438,180,100),(38,439,181,101),(39,440,182,102),(40,441,169,103),(41,442,170,104),(42,443,171,105),(43,391,71,403),(44,392,72,404),(45,379,73,405),(46,380,74,406),(47,381,75,393),(48,382,76,394),(49,383,77,395),(50,384,78,396),(51,385,79,397),(52,386,80,398),(53,387,81,399),(54,388,82,400),(55,389,83,401),(56,390,84,402),(57,293,312,305),(58,294,313,306),(59,281,314,307),(60,282,315,308),(61,283,316,295),(62,284,317,296),(63,285,318,297),(64,286,319,298),(65,287,320,299),(66,288,321,300),(67,289,322,301),(68,290,309,302),(69,291,310,303),(70,292,311,304),(85,198,346,273),(86,199,347,274),(87,200,348,275),(88,201,349,276),(89,202,350,277),(90,203,337,278),(91,204,338,279),(92,205,339,280),(93,206,340,267),(94,207,341,268),(95,208,342,269),(96,209,343,270),(97,210,344,271),(98,197,345,272),(113,430,407,225),(114,431,408,226),(115,432,409,227),(116,433,410,228),(117,434,411,229),(118,421,412,230),(119,422,413,231),(120,423,414,232),(121,424,415,233),(122,425,416,234),(123,426,417,235),(124,427,418,236),(125,428,419,237),(126,429,420,238),(127,330,368,217),(128,331,369,218),(129,332,370,219),(130,333,371,220),(131,334,372,221),(132,335,373,222),(133,336,374,223),(134,323,375,224),(135,324,376,211),(136,325,377,212),(137,326,378,213),(138,327,365,214),(139,328,366,215),(140,329,367,216)], [(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)], [(1,208,8,201),(2,207,9,200),(3,206,10,199),(4,205,11,198),(5,204,12,197),(6,203,13,210),(7,202,14,209),(15,404,22,397),(16,403,23,396),(17,402,24,395),(18,401,25,394),(19,400,26,393),(20,399,27,406),(21,398,28,405),(29,60,36,67),(30,59,37,66),(31,58,38,65),(32,57,39,64),(33,70,40,63),(34,69,41,62),(35,68,42,61),(43,263,50,256),(44,262,51,255),(45,261,52,254),(46,260,53,253),(47,259,54,266),(48,258,55,265),(49,257,56,264),(71,150,78,143),(72,149,79,142),(73,148,80,141),(74,147,81,154),(75,146,82,153),(76,145,83,152),(77,144,84,151),(85,186,92,193),(86,185,93,192),(87,184,94,191),(88,183,95,190),(89,196,96,189),(90,195,97,188),(91,194,98,187),(99,289,106,282),(100,288,107,281),(101,287,108,294),(102,286,109,293),(103,285,110,292),(104,284,111,291),(105,283,112,290),(113,220,120,213),(114,219,121,212),(115,218,122,211),(116,217,123,224),(117,216,124,223),(118,215,125,222),(119,214,126,221),(127,235,134,228),(128,234,135,227),(129,233,136,226),(130,232,137,225),(131,231,138,238),(132,230,139,237),(133,229,140,236),(155,350,162,343),(156,349,163,342),(157,348,164,341),(158,347,165,340),(159,346,166,339),(160,345,167,338),(161,344,168,337),(169,318,176,311),(170,317,177,310),(171,316,178,309),(172,315,179,322),(173,314,180,321),(174,313,181,320),(175,312,182,319),(239,390,246,383),(240,389,247,382),(241,388,248,381),(242,387,249,380),(243,386,250,379),(244,385,251,392),(245,384,252,391),(267,360,274,353),(268,359,275,352),(269,358,276,351),(270,357,277,364),(271,356,278,363),(272,355,279,362),(273,354,280,361),(295,436,302,443),(296,435,303,442),(297,448,304,441),(298,447,305,440),(299,446,306,439),(300,445,307,438),(301,444,308,437),(323,410,330,417),(324,409,331,416),(325,408,332,415),(326,407,333,414),(327,420,334,413),(328,419,335,412),(329,418,336,411),(365,429,372,422),(366,428,373,421),(367,427,374,434),(368,426,375,433),(369,425,376,432),(370,424,377,431),(371,423,378,430)]])
124 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D7 | Dic7 | D14 | Dic14 | D28 |
| kernel | C42⋊8Dic7 | C2×C4⋊Dic7 | C2×C4×C28 | C4×C28 | C2×C28 | C2×C28 | C2×C42 | C42 | C22×C4 | C2×C4 | C2×C4 |
| # reps | 1 | 6 | 1 | 8 | 6 | 6 | 3 | 12 | 9 | 36 | 36 |
Matrix representation of C42⋊8Dic7 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 24 | 0 | 0 |
| 0 | 5 | 13 | 0 | 0 |
| 0 | 0 | 0 | 2 | 11 |
| 0 | 0 | 0 | 18 | 27 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 2 | 11 |
| 0 | 0 | 0 | 18 | 27 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 1 | 11 | 0 | 0 |
| 0 | 0 | 0 | 26 | 28 |
| 0 | 0 | 0 | 1 | 0 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 9 |
| 0 | 0 | 0 | 7 | 9 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,16,5,0,0,0,24,13,0,0,0,0,0,2,18,0,0,0,11,27],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,2,18,0,0,0,11,27],[28,0,0,0,0,0,0,1,0,0,0,28,11,0,0,0,0,0,26,1,0,0,0,28,0],[17,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,20,7,0,0,0,9,9] >;
C42⋊8Dic7 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8{\rm Dic}_7 % in TeX
G:=Group("C4^2:8Dic7"); // GroupNames label
G:=SmallGroup(448,469);
// by ID
G=gap.SmallGroup(448,469);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,232,422,100,18822]);
// 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,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations