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G = (Q8×C14)⋊7C4  order 448 = 26·7

3rd semidirect product of Q8×C14 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (Q8×C14)⋊7C4, (C2×Dic7)⋊7Q8, (C2×Q8)⋊5Dic7, C14.28(C4×Q8), C2.6(Q8×Dic7), (C2×C28).195D4, C14.34(C4⋊Q8), (C22×Q8).6D7, C22.28(Q8×D7), C28.36(C22⋊C4), C2.5(D14⋊3Q8), (C22×C4).159D14, C14.80(C22⋊Q8), C4.13(C23.D7), C2.4(Dic7⋊Q8), C14.61(C4.4D4), C2.4(C28.23D4), C23.307(C22×D7), C14.C42.40C2, (C22×C28).398C22, (C22×C14).370C23, C7⋊4(C23.67C23), C22.31(Q8⋊2D7), C22.53(C22×Dic7), (C22×Dic7).198C22, (Q8×C2×C14).5C2, (C2×C14).88(C2×Q8), (C2×C4×Dic7).11C2, (C2×C28).189(C2×C4), (C2×C14).563(C2×D4), C14.80(C2×C22⋊C4), (C2×C4⋊Dic7).41C2, (C2×C4).27(C2×Dic7), C22.95(C2×C7⋊D4), C2.16(C2×C23.D7), (C2×C4).150(C7⋊D4), (C2×C14).194(C4○D4), (C2×C14).199(C22×C4), SmallGroup(448,764)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — (Q8×C14)⋊7C4
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4×Dic7 — (Q8×C14)⋊7C4
Lower central
C7 — C2×C14 — (Q8×C14)⋊7C4
Upper central
C1 — C23 — C22×Q8

Generators and relations for (Q8×C14)⋊7C4
 G = < a,b,c,d | a14=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, dbd-1=a7b-1, cd=dc >

Subgroups: 628 in 186 conjugacy classes, 91 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×C14, C23.67C23, C4×Dic7, C4⋊Dic7, C22×Dic7, C22×C28, C22×C28, Q8×C14, Q8×C14, C14.C42, C2×C4×Dic7, C2×C4⋊Dic7, Q8×C2×C14, (Q8×C14)⋊7C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, D14, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C2×Dic7, C7⋊D4, C22×D7, C23.67C23, C23.D7, Q8×D7, Q8⋊2D7, C22×Dic7, C2×C7⋊D4, Dic7⋊Q8, Q8×Dic7, D14⋊3Q8, C28.23D4, C2×C23.D7, (Q8×C14)⋊7C4

Smallest permutation representation of (Q8×C14)⋊7C4
►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)

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

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

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

88 conjugacy classes

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

88 irreducible representations

dim111111222222244
type+++++-+++--+
imageC1C2C2C2C2C4Q8D4D7C4○D4D14Dic7C7⋊D4Q8×D7Q8⋊2D7
kernel(Q8×C14)⋊7C4C14.C42C2×C4×Dic7C2×C4⋊Dic7Q8×C2×C14Q8×C14C2×Dic7C2×C28C22×Q8C2×C14C22×C4C2×Q8C2×C4C22C22
# reps14111844349122466

Matrix representation of (Q8×C14)⋊7C4 ►in GL7(𝔽29)

1000000
02400000
00230000
00028000
00002800
00000280
00000028
,
1000000
02800000
00280000
0001600
00002800
0000001
00000280
,
1000000
0100000
0010000
0001000
0000100
000002716
00000162
,
17000000
0010000
0100000
000112800
00061800
0000010
0000001

G:=sub<GL(7,GF(29))| [1,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,23,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,6,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,16,0,0,0,0,0,16,2],[17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,6,0,0,0,0,0,28,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
 

(Q8×C14)⋊7C4 in GAP, Magma, Sage, TeX 

(Q_8\times C_{14})\rtimes_7C_4
 
% in TeX
 
G:=Group("(Q8xC14):7C4");
 
// GroupNames label
 
G:=SmallGroup(448,764);
 
// by ID
 
G=gap.SmallGroup(448,764);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,387,184,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^14=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^7*b^-1,c*d=d*c>;
 
// generators/relations
 

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