metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊5Dic7, (C4×C28)⋊7C4, (C2×C42).6D7, C7⋊2(C42⋊5C4), (C22×C4).398D14, C2.2(C42⋊2D7), C14.4(C42⋊2C2), C22.46(C4○D28), C23.270(C22×D7), C14.43(C42⋊C2), C14.C42.13C2, (C22×C28).476C22, (C22×C14).312C23, C22.38(C22×Dic7), C2.7(C23.21D14), (C22×Dic7).31C22, (C2×C4×C28).3C2, (C2×C28).278(C2×C4), (C2×C4).63(C2×Dic7), (C2×C14).71(C4○D4), (C2×C14).177(C22×C4), SmallGroup(448,471)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊5Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 516 in 138 conjugacy classes, 71 normal (9 characteristic)
C1, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C42, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊5C4, C4×C28, C22×Dic7, C22×C28, C14.C42, C2×C4×C28, C42⋊5Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, Dic7, D14, C42⋊C2, C42⋊2C2, C2×Dic7, C22×D7, C42⋊5C4, C4○D28, C22×Dic7, C42⋊2D7, C23.21D14, C42⋊5Dic7
►Regular action on 448 points
Generators in S448
(1 237 249 113)(2 238 250 114)(3 225 251 115)(4 226 252 116)(5 227 239 117)(6 228 240 118)(7 229 241 119)(8 230 242 120)(9 231 243 121)(10 232 244 122)(11 233 245 123)(12 234 246 124)(13 235 247 125)(14 236 248 126)(15 164 388 181)(16 165 389 182)(17 166 390 169)(18 167 391 170)(19 168 392 171)(20 155 379 172)(21 156 380 173)(22 157 381 174)(23 158 382 175)(24 159 383 176)(25 160 384 177)(26 161 385 178)(27 162 386 179)(28 163 387 180)(29 266 198 326)(30 253 199 327)(31 254 200 328)(32 255 201 329)(33 256 202 330)(34 257 203 331)(35 258 204 332)(36 259 205 333)(37 260 206 334)(38 261 207 335)(39 262 208 336)(40 263 209 323)(41 264 210 324)(42 265 197 325)(43 71 97 430)(44 72 98 431)(45 73 85 432)(46 74 86 433)(47 75 87 434)(48 76 88 421)(49 77 89 422)(50 78 90 423)(51 79 91 424)(52 80 92 425)(53 81 93 426)(54 82 94 427)(55 83 95 428)(56 84 96 429)(57 409 374 186)(58 410 375 187)(59 411 376 188)(60 412 377 189)(61 413 378 190)(62 414 365 191)(63 415 366 192)(64 416 367 193)(65 417 368 194)(66 418 369 195)(67 419 370 196)(68 420 371 183)(69 407 372 184)(70 408 373 185)(99 403 290 275)(100 404 291 276)(101 405 292 277)(102 406 293 278)(103 393 294 279)(104 394 281 280)(105 395 282 267)(106 396 283 268)(107 397 284 269)(108 398 285 270)(109 399 286 271)(110 400 287 272)(111 401 288 273)(112 402 289 274)(127 302 363 348)(128 303 364 349)(129 304 351 350)(130 305 352 337)(131 306 353 338)(132 307 354 339)(133 308 355 340)(134 295 356 341)(135 296 357 342)(136 297 358 343)(137 298 359 344)(138 299 360 345)(139 300 361 346)(140 301 362 347)(141 213 441 310)(142 214 442 311)(143 215 443 312)(144 216 444 313)(145 217 445 314)(146 218 446 315)(147 219 447 316)(148 220 448 317)(149 221 435 318)(150 222 436 319)(151 223 437 320)(152 224 438 321)(153 211 439 322)(154 212 440 309)
(1 107 63 84)(2 108 64 71)(3 109 65 72)(4 110 66 73)(5 111 67 74)(6 112 68 75)(7 99 69 76)(8 100 70 77)(9 101 57 78)(10 102 58 79)(11 103 59 80)(12 104 60 81)(13 105 61 82)(14 106 62 83)(15 304 153 334)(16 305 154 335)(17 306 141 336)(18 307 142 323)(19 308 143 324)(20 295 144 325)(21 296 145 326)(22 297 146 327)(23 298 147 328)(24 299 148 329)(25 300 149 330)(26 301 150 331)(27 302 151 332)(28 303 152 333)(29 156 357 217)(30 157 358 218)(31 158 359 219)(32 159 360 220)(33 160 361 221)(34 161 362 222)(35 162 363 223)(36 163 364 224)(37 164 351 211)(38 165 352 212)(39 166 353 213)(40 167 354 214)(41 168 355 215)(42 155 356 216)(43 114 270 193)(44 115 271 194)(45 116 272 195)(46 117 273 196)(47 118 274 183)(48 119 275 184)(49 120 276 185)(50 121 277 186)(51 122 278 187)(52 123 279 188)(53 124 280 189)(54 125 267 190)(55 126 268 191)(56 113 269 192)(85 226 400 418)(86 227 401 419)(87 228 402 420)(88 229 403 407)(89 230 404 408)(90 231 405 409)(91 232 406 410)(92 233 393 411)(93 234 394 412)(94 235 395 413)(95 236 396 414)(96 237 397 415)(97 238 398 416)(98 225 399 417)(127 320 204 179)(128 321 205 180)(129 322 206 181)(130 309 207 182)(131 310 208 169)(132 311 209 170)(133 312 210 171)(134 313 197 172)(135 314 198 173)(136 315 199 174)(137 316 200 175)(138 317 201 176)(139 318 202 177)(140 319 203 178)(239 288 370 433)(240 289 371 434)(241 290 372 421)(242 291 373 422)(243 292 374 423)(244 293 375 424)(245 294 376 425)(246 281 377 426)(247 282 378 427)(248 283 365 428)(249 284 366 429)(250 285 367 430)(251 286 368 431)(252 287 369 432)(253 381 343 446)(254 382 344 447)(255 383 345 448)(256 384 346 435)(257 385 347 436)(258 386 348 437)(259 387 349 438)(260 388 350 439)(261 389 337 440)(262 390 338 441)(263 391 339 442)(264 392 340 443)(265 379 341 444)(266 380 342 445)
(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)
(1 131 8 138)(2 130 9 137)(3 129 10 136)(4 128 11 135)(5 127 12 134)(6 140 13 133)(7 139 14 132)(15 51 22 44)(16 50 23 43)(17 49 24 56)(18 48 25 55)(19 47 26 54)(20 46 27 53)(21 45 28 52)(29 369 36 376)(30 368 37 375)(31 367 38 374)(32 366 39 373)(33 365 40 372)(34 378 41 371)(35 377 42 370)(57 200 64 207)(58 199 65 206)(59 198 66 205)(60 197 67 204)(61 210 68 203)(62 209 69 202)(63 208 70 201)(71 212 78 219)(72 211 79 218)(73 224 80 217)(74 223 81 216)(75 222 82 215)(76 221 83 214)(77 220 84 213)(85 387 92 380)(86 386 93 379)(87 385 94 392)(88 384 95 391)(89 383 96 390)(90 382 97 389)(91 381 98 388)(99 160 106 167)(100 159 107 166)(101 158 108 165)(102 157 109 164)(103 156 110 163)(104 155 111 162)(105 168 112 161)(113 262 120 255)(114 261 121 254)(115 260 122 253)(116 259 123 266)(117 258 124 265)(118 257 125 264)(119 256 126 263)(141 276 148 269)(142 275 149 268)(143 274 150 267)(144 273 151 280)(145 272 152 279)(146 271 153 278)(147 270 154 277)(169 291 176 284)(170 290 177 283)(171 289 178 282)(172 288 179 281)(173 287 180 294)(174 286 181 293)(175 285 182 292)(183 347 190 340)(184 346 191 339)(185 345 192 338)(186 344 193 337)(187 343 194 350)(188 342 195 349)(189 341 196 348)(225 334 232 327)(226 333 233 326)(227 332 234 325)(228 331 235 324)(229 330 236 323)(230 329 237 336)(231 328 238 335)(239 363 246 356)(240 362 247 355)(241 361 248 354)(242 360 249 353)(243 359 250 352)(244 358 251 351)(245 357 252 364)(295 419 302 412)(296 418 303 411)(297 417 304 410)(298 416 305 409)(299 415 306 408)(300 414 307 407)(301 413 308 420)(309 423 316 430)(310 422 317 429)(311 421 318 428)(312 434 319 427)(313 433 320 426)(314 432 321 425)(315 431 322 424)(393 445 400 438)(394 444 401 437)(395 443 402 436)(396 442 403 435)(397 441 404 448)(398 440 405 447)(399 439 406 446)
G:=sub<Sym(448)| (1,237,249,113)(2,238,250,114)(3,225,251,115)(4,226,252,116)(5,227,239,117)(6,228,240,118)(7,229,241,119)(8,230,242,120)(9,231,243,121)(10,232,244,122)(11,233,245,123)(12,234,246,124)(13,235,247,125)(14,236,248,126)(15,164,388,181)(16,165,389,182)(17,166,390,169)(18,167,391,170)(19,168,392,171)(20,155,379,172)(21,156,380,173)(22,157,381,174)(23,158,382,175)(24,159,383,176)(25,160,384,177)(26,161,385,178)(27,162,386,179)(28,163,387,180)(29,266,198,326)(30,253,199,327)(31,254,200,328)(32,255,201,329)(33,256,202,330)(34,257,203,331)(35,258,204,332)(36,259,205,333)(37,260,206,334)(38,261,207,335)(39,262,208,336)(40,263,209,323)(41,264,210,324)(42,265,197,325)(43,71,97,430)(44,72,98,431)(45,73,85,432)(46,74,86,433)(47,75,87,434)(48,76,88,421)(49,77,89,422)(50,78,90,423)(51,79,91,424)(52,80,92,425)(53,81,93,426)(54,82,94,427)(55,83,95,428)(56,84,96,429)(57,409,374,186)(58,410,375,187)(59,411,376,188)(60,412,377,189)(61,413,378,190)(62,414,365,191)(63,415,366,192)(64,416,367,193)(65,417,368,194)(66,418,369,195)(67,419,370,196)(68,420,371,183)(69,407,372,184)(70,408,373,185)(99,403,290,275)(100,404,291,276)(101,405,292,277)(102,406,293,278)(103,393,294,279)(104,394,281,280)(105,395,282,267)(106,396,283,268)(107,397,284,269)(108,398,285,270)(109,399,286,271)(110,400,287,272)(111,401,288,273)(112,402,289,274)(127,302,363,348)(128,303,364,349)(129,304,351,350)(130,305,352,337)(131,306,353,338)(132,307,354,339)(133,308,355,340)(134,295,356,341)(135,296,357,342)(136,297,358,343)(137,298,359,344)(138,299,360,345)(139,300,361,346)(140,301,362,347)(141,213,441,310)(142,214,442,311)(143,215,443,312)(144,216,444,313)(145,217,445,314)(146,218,446,315)(147,219,447,316)(148,220,448,317)(149,221,435,318)(150,222,436,319)(151,223,437,320)(152,224,438,321)(153,211,439,322)(154,212,440,309), (1,107,63,84)(2,108,64,71)(3,109,65,72)(4,110,66,73)(5,111,67,74)(6,112,68,75)(7,99,69,76)(8,100,70,77)(9,101,57,78)(10,102,58,79)(11,103,59,80)(12,104,60,81)(13,105,61,82)(14,106,62,83)(15,304,153,334)(16,305,154,335)(17,306,141,336)(18,307,142,323)(19,308,143,324)(20,295,144,325)(21,296,145,326)(22,297,146,327)(23,298,147,328)(24,299,148,329)(25,300,149,330)(26,301,150,331)(27,302,151,332)(28,303,152,333)(29,156,357,217)(30,157,358,218)(31,158,359,219)(32,159,360,220)(33,160,361,221)(34,161,362,222)(35,162,363,223)(36,163,364,224)(37,164,351,211)(38,165,352,212)(39,166,353,213)(40,167,354,214)(41,168,355,215)(42,155,356,216)(43,114,270,193)(44,115,271,194)(45,116,272,195)(46,117,273,196)(47,118,274,183)(48,119,275,184)(49,120,276,185)(50,121,277,186)(51,122,278,187)(52,123,279,188)(53,124,280,189)(54,125,267,190)(55,126,268,191)(56,113,269,192)(85,226,400,418)(86,227,401,419)(87,228,402,420)(88,229,403,407)(89,230,404,408)(90,231,405,409)(91,232,406,410)(92,233,393,411)(93,234,394,412)(94,235,395,413)(95,236,396,414)(96,237,397,415)(97,238,398,416)(98,225,399,417)(127,320,204,179)(128,321,205,180)(129,322,206,181)(130,309,207,182)(131,310,208,169)(132,311,209,170)(133,312,210,171)(134,313,197,172)(135,314,198,173)(136,315,199,174)(137,316,200,175)(138,317,201,176)(139,318,202,177)(140,319,203,178)(239,288,370,433)(240,289,371,434)(241,290,372,421)(242,291,373,422)(243,292,374,423)(244,293,375,424)(245,294,376,425)(246,281,377,426)(247,282,378,427)(248,283,365,428)(249,284,366,429)(250,285,367,430)(251,286,368,431)(252,287,369,432)(253,381,343,446)(254,382,344,447)(255,383,345,448)(256,384,346,435)(257,385,347,436)(258,386,348,437)(259,387,349,438)(260,388,350,439)(261,389,337,440)(262,390,338,441)(263,391,339,442)(264,392,340,443)(265,379,341,444)(266,380,342,445), (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), (1,131,8,138)(2,130,9,137)(3,129,10,136)(4,128,11,135)(5,127,12,134)(6,140,13,133)(7,139,14,132)(15,51,22,44)(16,50,23,43)(17,49,24,56)(18,48,25,55)(19,47,26,54)(20,46,27,53)(21,45,28,52)(29,369,36,376)(30,368,37,375)(31,367,38,374)(32,366,39,373)(33,365,40,372)(34,378,41,371)(35,377,42,370)(57,200,64,207)(58,199,65,206)(59,198,66,205)(60,197,67,204)(61,210,68,203)(62,209,69,202)(63,208,70,201)(71,212,78,219)(72,211,79,218)(73,224,80,217)(74,223,81,216)(75,222,82,215)(76,221,83,214)(77,220,84,213)(85,387,92,380)(86,386,93,379)(87,385,94,392)(88,384,95,391)(89,383,96,390)(90,382,97,389)(91,381,98,388)(99,160,106,167)(100,159,107,166)(101,158,108,165)(102,157,109,164)(103,156,110,163)(104,155,111,162)(105,168,112,161)(113,262,120,255)(114,261,121,254)(115,260,122,253)(116,259,123,266)(117,258,124,265)(118,257,125,264)(119,256,126,263)(141,276,148,269)(142,275,149,268)(143,274,150,267)(144,273,151,280)(145,272,152,279)(146,271,153,278)(147,270,154,277)(169,291,176,284)(170,290,177,283)(171,289,178,282)(172,288,179,281)(173,287,180,294)(174,286,181,293)(175,285,182,292)(183,347,190,340)(184,346,191,339)(185,345,192,338)(186,344,193,337)(187,343,194,350)(188,342,195,349)(189,341,196,348)(225,334,232,327)(226,333,233,326)(227,332,234,325)(228,331,235,324)(229,330,236,323)(230,329,237,336)(231,328,238,335)(239,363,246,356)(240,362,247,355)(241,361,248,354)(242,360,249,353)(243,359,250,352)(244,358,251,351)(245,357,252,364)(295,419,302,412)(296,418,303,411)(297,417,304,410)(298,416,305,409)(299,415,306,408)(300,414,307,407)(301,413,308,420)(309,423,316,430)(310,422,317,429)(311,421,318,428)(312,434,319,427)(313,433,320,426)(314,432,321,425)(315,431,322,424)(393,445,400,438)(394,444,401,437)(395,443,402,436)(396,442,403,435)(397,441,404,448)(398,440,405,447)(399,439,406,446)>;
G:=Group( (1,237,249,113)(2,238,250,114)(3,225,251,115)(4,226,252,116)(5,227,239,117)(6,228,240,118)(7,229,241,119)(8,230,242,120)(9,231,243,121)(10,232,244,122)(11,233,245,123)(12,234,246,124)(13,235,247,125)(14,236,248,126)(15,164,388,181)(16,165,389,182)(17,166,390,169)(18,167,391,170)(19,168,392,171)(20,155,379,172)(21,156,380,173)(22,157,381,174)(23,158,382,175)(24,159,383,176)(25,160,384,177)(26,161,385,178)(27,162,386,179)(28,163,387,180)(29,266,198,326)(30,253,199,327)(31,254,200,328)(32,255,201,329)(33,256,202,330)(34,257,203,331)(35,258,204,332)(36,259,205,333)(37,260,206,334)(38,261,207,335)(39,262,208,336)(40,263,209,323)(41,264,210,324)(42,265,197,325)(43,71,97,430)(44,72,98,431)(45,73,85,432)(46,74,86,433)(47,75,87,434)(48,76,88,421)(49,77,89,422)(50,78,90,423)(51,79,91,424)(52,80,92,425)(53,81,93,426)(54,82,94,427)(55,83,95,428)(56,84,96,429)(57,409,374,186)(58,410,375,187)(59,411,376,188)(60,412,377,189)(61,413,378,190)(62,414,365,191)(63,415,366,192)(64,416,367,193)(65,417,368,194)(66,418,369,195)(67,419,370,196)(68,420,371,183)(69,407,372,184)(70,408,373,185)(99,403,290,275)(100,404,291,276)(101,405,292,277)(102,406,293,278)(103,393,294,279)(104,394,281,280)(105,395,282,267)(106,396,283,268)(107,397,284,269)(108,398,285,270)(109,399,286,271)(110,400,287,272)(111,401,288,273)(112,402,289,274)(127,302,363,348)(128,303,364,349)(129,304,351,350)(130,305,352,337)(131,306,353,338)(132,307,354,339)(133,308,355,340)(134,295,356,341)(135,296,357,342)(136,297,358,343)(137,298,359,344)(138,299,360,345)(139,300,361,346)(140,301,362,347)(141,213,441,310)(142,214,442,311)(143,215,443,312)(144,216,444,313)(145,217,445,314)(146,218,446,315)(147,219,447,316)(148,220,448,317)(149,221,435,318)(150,222,436,319)(151,223,437,320)(152,224,438,321)(153,211,439,322)(154,212,440,309), (1,107,63,84)(2,108,64,71)(3,109,65,72)(4,110,66,73)(5,111,67,74)(6,112,68,75)(7,99,69,76)(8,100,70,77)(9,101,57,78)(10,102,58,79)(11,103,59,80)(12,104,60,81)(13,105,61,82)(14,106,62,83)(15,304,153,334)(16,305,154,335)(17,306,141,336)(18,307,142,323)(19,308,143,324)(20,295,144,325)(21,296,145,326)(22,297,146,327)(23,298,147,328)(24,299,148,329)(25,300,149,330)(26,301,150,331)(27,302,151,332)(28,303,152,333)(29,156,357,217)(30,157,358,218)(31,158,359,219)(32,159,360,220)(33,160,361,221)(34,161,362,222)(35,162,363,223)(36,163,364,224)(37,164,351,211)(38,165,352,212)(39,166,353,213)(40,167,354,214)(41,168,355,215)(42,155,356,216)(43,114,270,193)(44,115,271,194)(45,116,272,195)(46,117,273,196)(47,118,274,183)(48,119,275,184)(49,120,276,185)(50,121,277,186)(51,122,278,187)(52,123,279,188)(53,124,280,189)(54,125,267,190)(55,126,268,191)(56,113,269,192)(85,226,400,418)(86,227,401,419)(87,228,402,420)(88,229,403,407)(89,230,404,408)(90,231,405,409)(91,232,406,410)(92,233,393,411)(93,234,394,412)(94,235,395,413)(95,236,396,414)(96,237,397,415)(97,238,398,416)(98,225,399,417)(127,320,204,179)(128,321,205,180)(129,322,206,181)(130,309,207,182)(131,310,208,169)(132,311,209,170)(133,312,210,171)(134,313,197,172)(135,314,198,173)(136,315,199,174)(137,316,200,175)(138,317,201,176)(139,318,202,177)(140,319,203,178)(239,288,370,433)(240,289,371,434)(241,290,372,421)(242,291,373,422)(243,292,374,423)(244,293,375,424)(245,294,376,425)(246,281,377,426)(247,282,378,427)(248,283,365,428)(249,284,366,429)(250,285,367,430)(251,286,368,431)(252,287,369,432)(253,381,343,446)(254,382,344,447)(255,383,345,448)(256,384,346,435)(257,385,347,436)(258,386,348,437)(259,387,349,438)(260,388,350,439)(261,389,337,440)(262,390,338,441)(263,391,339,442)(264,392,340,443)(265,379,341,444)(266,380,342,445), (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), (1,131,8,138)(2,130,9,137)(3,129,10,136)(4,128,11,135)(5,127,12,134)(6,140,13,133)(7,139,14,132)(15,51,22,44)(16,50,23,43)(17,49,24,56)(18,48,25,55)(19,47,26,54)(20,46,27,53)(21,45,28,52)(29,369,36,376)(30,368,37,375)(31,367,38,374)(32,366,39,373)(33,365,40,372)(34,378,41,371)(35,377,42,370)(57,200,64,207)(58,199,65,206)(59,198,66,205)(60,197,67,204)(61,210,68,203)(62,209,69,202)(63,208,70,201)(71,212,78,219)(72,211,79,218)(73,224,80,217)(74,223,81,216)(75,222,82,215)(76,221,83,214)(77,220,84,213)(85,387,92,380)(86,386,93,379)(87,385,94,392)(88,384,95,391)(89,383,96,390)(90,382,97,389)(91,381,98,388)(99,160,106,167)(100,159,107,166)(101,158,108,165)(102,157,109,164)(103,156,110,163)(104,155,111,162)(105,168,112,161)(113,262,120,255)(114,261,121,254)(115,260,122,253)(116,259,123,266)(117,258,124,265)(118,257,125,264)(119,256,126,263)(141,276,148,269)(142,275,149,268)(143,274,150,267)(144,273,151,280)(145,272,152,279)(146,271,153,278)(147,270,154,277)(169,291,176,284)(170,290,177,283)(171,289,178,282)(172,288,179,281)(173,287,180,294)(174,286,181,293)(175,285,182,292)(183,347,190,340)(184,346,191,339)(185,345,192,338)(186,344,193,337)(187,343,194,350)(188,342,195,349)(189,341,196,348)(225,334,232,327)(226,333,233,326)(227,332,234,325)(228,331,235,324)(229,330,236,323)(230,329,237,336)(231,328,238,335)(239,363,246,356)(240,362,247,355)(241,361,248,354)(242,360,249,353)(243,359,250,352)(244,358,251,351)(245,357,252,364)(295,419,302,412)(296,418,303,411)(297,417,304,410)(298,416,305,409)(299,415,306,408)(300,414,307,407)(301,413,308,420)(309,423,316,430)(310,422,317,429)(311,421,318,428)(312,434,319,427)(313,433,320,426)(314,432,321,425)(315,431,322,424)(393,445,400,438)(394,444,401,437)(395,443,402,436)(396,442,403,435)(397,441,404,448)(398,440,405,447)(399,439,406,446) );
G=PermutationGroup([[(1,237,249,113),(2,238,250,114),(3,225,251,115),(4,226,252,116),(5,227,239,117),(6,228,240,118),(7,229,241,119),(8,230,242,120),(9,231,243,121),(10,232,244,122),(11,233,245,123),(12,234,246,124),(13,235,247,125),(14,236,248,126),(15,164,388,181),(16,165,389,182),(17,166,390,169),(18,167,391,170),(19,168,392,171),(20,155,379,172),(21,156,380,173),(22,157,381,174),(23,158,382,175),(24,159,383,176),(25,160,384,177),(26,161,385,178),(27,162,386,179),(28,163,387,180),(29,266,198,326),(30,253,199,327),(31,254,200,328),(32,255,201,329),(33,256,202,330),(34,257,203,331),(35,258,204,332),(36,259,205,333),(37,260,206,334),(38,261,207,335),(39,262,208,336),(40,263,209,323),(41,264,210,324),(42,265,197,325),(43,71,97,430),(44,72,98,431),(45,73,85,432),(46,74,86,433),(47,75,87,434),(48,76,88,421),(49,77,89,422),(50,78,90,423),(51,79,91,424),(52,80,92,425),(53,81,93,426),(54,82,94,427),(55,83,95,428),(56,84,96,429),(57,409,374,186),(58,410,375,187),(59,411,376,188),(60,412,377,189),(61,413,378,190),(62,414,365,191),(63,415,366,192),(64,416,367,193),(65,417,368,194),(66,418,369,195),(67,419,370,196),(68,420,371,183),(69,407,372,184),(70,408,373,185),(99,403,290,275),(100,404,291,276),(101,405,292,277),(102,406,293,278),(103,393,294,279),(104,394,281,280),(105,395,282,267),(106,396,283,268),(107,397,284,269),(108,398,285,270),(109,399,286,271),(110,400,287,272),(111,401,288,273),(112,402,289,274),(127,302,363,348),(128,303,364,349),(129,304,351,350),(130,305,352,337),(131,306,353,338),(132,307,354,339),(133,308,355,340),(134,295,356,341),(135,296,357,342),(136,297,358,343),(137,298,359,344),(138,299,360,345),(139,300,361,346),(140,301,362,347),(141,213,441,310),(142,214,442,311),(143,215,443,312),(144,216,444,313),(145,217,445,314),(146,218,446,315),(147,219,447,316),(148,220,448,317),(149,221,435,318),(150,222,436,319),(151,223,437,320),(152,224,438,321),(153,211,439,322),(154,212,440,309)], [(1,107,63,84),(2,108,64,71),(3,109,65,72),(4,110,66,73),(5,111,67,74),(6,112,68,75),(7,99,69,76),(8,100,70,77),(9,101,57,78),(10,102,58,79),(11,103,59,80),(12,104,60,81),(13,105,61,82),(14,106,62,83),(15,304,153,334),(16,305,154,335),(17,306,141,336),(18,307,142,323),(19,308,143,324),(20,295,144,325),(21,296,145,326),(22,297,146,327),(23,298,147,328),(24,299,148,329),(25,300,149,330),(26,301,150,331),(27,302,151,332),(28,303,152,333),(29,156,357,217),(30,157,358,218),(31,158,359,219),(32,159,360,220),(33,160,361,221),(34,161,362,222),(35,162,363,223),(36,163,364,224),(37,164,351,211),(38,165,352,212),(39,166,353,213),(40,167,354,214),(41,168,355,215),(42,155,356,216),(43,114,270,193),(44,115,271,194),(45,116,272,195),(46,117,273,196),(47,118,274,183),(48,119,275,184),(49,120,276,185),(50,121,277,186),(51,122,278,187),(52,123,279,188),(53,124,280,189),(54,125,267,190),(55,126,268,191),(56,113,269,192),(85,226,400,418),(86,227,401,419),(87,228,402,420),(88,229,403,407),(89,230,404,408),(90,231,405,409),(91,232,406,410),(92,233,393,411),(93,234,394,412),(94,235,395,413),(95,236,396,414),(96,237,397,415),(97,238,398,416),(98,225,399,417),(127,320,204,179),(128,321,205,180),(129,322,206,181),(130,309,207,182),(131,310,208,169),(132,311,209,170),(133,312,210,171),(134,313,197,172),(135,314,198,173),(136,315,199,174),(137,316,200,175),(138,317,201,176),(139,318,202,177),(140,319,203,178),(239,288,370,433),(240,289,371,434),(241,290,372,421),(242,291,373,422),(243,292,374,423),(244,293,375,424),(245,294,376,425),(246,281,377,426),(247,282,378,427),(248,283,365,428),(249,284,366,429),(250,285,367,430),(251,286,368,431),(252,287,369,432),(253,381,343,446),(254,382,344,447),(255,383,345,448),(256,384,346,435),(257,385,347,436),(258,386,348,437),(259,387,349,438),(260,388,350,439),(261,389,337,440),(262,390,338,441),(263,391,339,442),(264,392,340,443),(265,379,341,444),(266,380,342,445)], [(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)], [(1,131,8,138),(2,130,9,137),(3,129,10,136),(4,128,11,135),(5,127,12,134),(6,140,13,133),(7,139,14,132),(15,51,22,44),(16,50,23,43),(17,49,24,56),(18,48,25,55),(19,47,26,54),(20,46,27,53),(21,45,28,52),(29,369,36,376),(30,368,37,375),(31,367,38,374),(32,366,39,373),(33,365,40,372),(34,378,41,371),(35,377,42,370),(57,200,64,207),(58,199,65,206),(59,198,66,205),(60,197,67,204),(61,210,68,203),(62,209,69,202),(63,208,70,201),(71,212,78,219),(72,211,79,218),(73,224,80,217),(74,223,81,216),(75,222,82,215),(76,221,83,214),(77,220,84,213),(85,387,92,380),(86,386,93,379),(87,385,94,392),(88,384,95,391),(89,383,96,390),(90,382,97,389),(91,381,98,388),(99,160,106,167),(100,159,107,166),(101,158,108,165),(102,157,109,164),(103,156,110,163),(104,155,111,162),(105,168,112,161),(113,262,120,255),(114,261,121,254),(115,260,122,253),(116,259,123,266),(117,258,124,265),(118,257,125,264),(119,256,126,263),(141,276,148,269),(142,275,149,268),(143,274,150,267),(144,273,151,280),(145,272,152,279),(146,271,153,278),(147,270,154,277),(169,291,176,284),(170,290,177,283),(171,289,178,282),(172,288,179,281),(173,287,180,294),(174,286,181,293),(175,285,182,292),(183,347,190,340),(184,346,191,339),(185,345,192,338),(186,344,193,337),(187,343,194,350),(188,342,195,349),(189,341,196,348),(225,334,232,327),(226,333,233,326),(227,332,234,325),(228,331,235,324),(229,330,236,323),(230,329,237,336),(231,328,238,335),(239,363,246,356),(240,362,247,355),(241,361,248,354),(242,360,249,353),(243,359,250,352),(244,358,251,351),(245,357,252,364),(295,419,302,412),(296,418,303,411),(297,417,304,410),(298,416,305,409),(299,415,306,408),(300,414,307,407),(301,413,308,420),(309,423,316,430),(310,422,317,429),(311,421,318,428),(312,434,319,427),(313,433,320,426),(314,432,321,425),(315,431,322,424),(393,445,400,438),(394,444,401,437),(395,443,402,436),(396,442,403,435),(397,441,404,448),(398,440,405,447),(399,439,406,446)]])
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 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | D7 | C4○D4 | Dic7 | D14 | C4○D28 |
| kernel | C42⋊5Dic7 | C14.C42 | C2×C4×C28 | C4×C28 | C2×C42 | C2×C14 | C42 | C22×C4 | C22 |
| # reps | 1 | 6 | 1 | 8 | 3 | 12 | 12 | 9 | 72 |
Matrix representation of C42⋊5Dic7 ►in GL6(𝔽29)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 18 |
| 0 | 0 | 0 | 0 | 27 | 21 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 28 | 0 | 0 |
| 0 | 0 | 20 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 1 |
| 0 | 0 | 0 | 0 | 16 | 22 |
| 9 | 6 | 0 | 0 | 0 | 0 |
| 6 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 16 | 0 | 0 |
| 0 | 0 | 8 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 17 |
| 0 | 0 | 0 | 0 | 26 | 9 |
G:=sub<GL(6,GF(29))| [0,17,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,27,0,0,0,0,18,21],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,28,10,0,0,0,0,0,0,10,16,0,0,0,0,1,22],[9,6,0,0,0,0,6,20,0,0,0,0,0,0,25,8,0,0,0,0,16,4,0,0,0,0,0,0,20,26,0,0,0,0,17,9] >;
C42⋊5Dic7 in GAP, Magma, Sage, TeX
C_4^2\rtimes_5{\rm Dic}_7 % in TeX
G:=Group("C4^2:5Dic7"); // GroupNames label
G:=SmallGroup(448,471);
// by ID
G=gap.SmallGroup(448,471);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,120,1094,184,18822]);
// 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*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations