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G = C7⋊(C42⋊5C4)  order 448 = 26·7

The semidirect product of C7 and C42⋊5C4 acting via C42⋊5C4/C2.C42=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C7⋊1(C42⋊5C4), (C4×Dic7)⋊13C4, (C22×C4).8D14, C2.6(C42⋊D7), (C22×C28).4C22, C2.C42.2D7, C14.6(C42⋊2C2), C14.2(C42⋊C2), C22.28(C4○D28), C14.C42.2C2, C23.248(C22×D7), C22.30(D4⋊2D7), (C22×C14).276C23, C22.13(Q8⋊2D7), C2.1(C23.D14), C2.6(C23.11D14), (C22×Dic7).172C22, C22.83(C2×C4×D7), (C2×C4).123(C4×D7), (C2×C4×Dic7).24C2, (C2×C28).140(C2×C4), C2.5(C4⋊C4⋊7D7), C2.1(C4⋊C4⋊D7), (C2×C14).42(C22×C4), (C2×Dic7).76(C2×C4), (C2×C14).124(C4○D4), (C7×C2.C42).22C2, SmallGroup(448,185)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C7⋊(C42⋊5C4)
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4×Dic7 — C7⋊(C42⋊5C4)
Lower central
C7 — C2×C14 — C7⋊(C42⋊5C4)
Upper central
C1 — C23 — C2.C42

Generators and relations for C7⋊(C42⋊5C4)
 G = < a,b,c,d | a7=b4=c4=d4=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c-1 >

Subgroups: 540 in 138 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C22×C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2.C42, C2×C42, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C42⋊5C4, C4×Dic7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C14.C42, C7×C2.C42, C2×C4×Dic7, C7⋊(C42⋊5C4)
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C42⋊2C2, C4×D7, C22×D7, C42⋊5C4, C2×C4×D7, C4○D28, D4⋊2D7, Q8⋊2D7, C42⋊D7, C23.11D14, C23.D14, C4⋊C4⋊7D7, C4⋊C4⋊D7, C7⋊(C42⋊5C4)

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

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

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

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

88 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T7A7B7C14A···14U28A···28AJ
order12···2444444444···4444477714···1428···28
size11···12222444414···14282828282222···24···4

88 irreducible representations

dim111112222244
type++++++-+
imageC1C2C2C2C4D7C4○D4D14C4×D7C4○D28D4⋊2D7Q8⋊2D7
kernelC7⋊(C42⋊5C4)C14.C42C7×C2.C42C2×C4×Dic7C4×Dic7C2.C42C2×C14C22×C4C2×C4C22C22C22
# reps151183129122493

Matrix representation of C7⋊(C42⋊5C4) ►in GL7(𝔽29)

1000000
0100000
0010000
0001000
0000100
00000160
00000020
,
28000000
01200000
00120000
00014500
00071500
0000001
0000010
,
1000000
07160000
026220000
00012000
00001200
0000010
0000001
,
12000000
012270000
00170000
000122800
000271700
0000010
0000001

G:=sub<GL(7,GF(29))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,20],[28,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,14,7,0,0,0,0,0,5,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,7,26,0,0,0,0,0,16,22,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,27,17,0,0,0,0,0,0,0,12,27,0,0,0,0,0,28,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
 

C7⋊(C42⋊5C4) in GAP, Magma, Sage, TeX 

C_7\rtimes (C_4^2\rtimes_5C_4)
 
% in TeX
 
G:=Group("C7:(C4^2:5C4)");
 
// GroupNames label
 
G:=SmallGroup(448,185);
 
// by ID
 
G=gap.SmallGroup(448,185);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,1094,387,58,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

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