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G = C7×C23.63C23  order 448 = 26·7

Direct product of C7 and C23.63C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C23.63C23, C4⋊C4⋊5C28, C2.6(D4×C28), C2.3(Q8×C28), (C2×C28).50Q8, C14.30(C4×Q8), C14.107(C4×D4), (C2×C28).455D4, (C2×C42).5C14, C22.36(D4×C14), C22.13(Q8×C14), C14.84(C22⋊Q8), C2.C42.3C14, (C22×C28).32C22, C22.36(C22×C28), C23.60(C22×C14), C14.27(C42.C2), C14.32(C42⋊2C2), C14.57(C42⋊C2), (C22×C14).451C23, C14.88(C22.D4), (C2×C4×C28).7C2, (C7×C4⋊C4)⋊12C4, (C2×C4⋊C4).6C14, (C14×C4⋊C4).35C2, (C2×C4).10(C7×Q8), (C2×C4).33(C2×C28), C2.3(C7×C22⋊Q8), (C2×C4).100(C7×D4), (C2×C28).190(C2×C4), C2.2(C7×C42.C2), (C2×C14).603(C2×D4), C2.9(C7×C42⋊C2), C2.2(C7×C42⋊2C2), (C2×C14).105(C2×Q8), C22.21(C7×C4○D4), (C22×C4).20(C2×C14), (C2×C14).211(C4○D4), C2.4(C7×C22.D4), (C2×C14).223(C22×C4), (C7×C2.C42).5C2, SmallGroup(448,795)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C23.63C23
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C4⋊C4 — C7×C23.63C23
Lower central
C1 — C22 — C7×C23.63C23
Upper central
C1 — C22×C14 — C7×C23.63C23

Generators and relations for C7×C23.63C23
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=f2=d, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 226 in 154 conjugacy classes, 90 normal (62 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C4⋊C4, C22×C4, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C23.63C23, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C22×C28, C7×C2.C42, C2×C4×C28, C14×C4⋊C4, C7×C23.63C23
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×C28, C7×D4, C7×Q8, C22×C14, C23.63C23, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C7×C42⋊C2, D4×C28, Q8×C28, C7×C22⋊Q8, C7×C22.D4, C7×C42.C2, C7×C42⋊2C2, C7×C23.63C23

Smallest permutation representation of C7×C23.63C23
►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 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 441)(16 435)(17 436)(18 437)(19 438)(20 439)(21 440)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(36 446)(37 447)(38 448)(39 442)(40 443)(41 444)(42 445)(50 77)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(64 84)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(85 95)(86 96)(87 97)(88 98)(89 92)(90 93)(91 94)(99 143)(100 144)(101 145)(102 146)(103 147)(104 141)(105 142)(106 133)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 148)(120 140)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(155 199)(156 200)(157 201)(158 202)(159 203)(160 197)(161 198)(162 189)(163 183)(164 184)(165 185)(166 186)(167 187)(168 188)(169 205)(170 206)(171 207)(172 208)(173 209)(174 210)(175 204)(176 196)(177 190)(178 191)(179 192)(180 193)(181 194)(182 195)(211 255)(212 256)(213 257)(214 258)(215 259)(216 253)(217 254)(218 245)(219 239)(220 240)(221 241)(222 242)(223 243)(224 244)(225 261)(226 262)(227 263)(228 264)(229 265)(230 266)(231 260)(232 252)(233 246)(234 247)(235 248)(236 249)(237 250)(238 251)(267 311)(268 312)(269 313)(270 314)(271 315)(272 309)(273 310)(274 301)(275 295)(276 296)(277 297)(278 298)(279 299)(280 300)(281 317)(282 318)(283 319)(284 320)(285 321)(286 322)(287 316)(288 308)(289 302)(290 303)(291 304)(292 305)(293 306)(294 307)(323 367)(324 368)(325 369)(326 370)(327 371)(328 365)(329 366)(330 357)(331 351)(332 352)(333 353)(334 354)(335 355)(336 356)(337 373)(338 374)(339 375)(340 376)(341 377)(342 378)(343 372)(344 364)(345 358)(346 359)(347 360)(348 361)(349 362)(350 363)(379 423)(380 424)(381 425)(382 426)(383 427)(384 421)(385 422)(386 413)(387 407)(388 408)(389 409)(390 410)(391 411)(392 412)(393 429)(394 430)(395 431)(396 432)(397 433)(398 434)(399 428)(400 420)(401 414)(402 415)(403 416)(404 417)(405 418)(406 419)

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

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

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

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

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

196 conjugacy classes

class 1 2A···2G4A···4L4M···4T7A···7F14A···14AP28A···28BT28BU···28DP
order12···24···44···47···714···1428···2828···28
size11···12···24···41···11···12···24···4

196 irreducible representations

dim1111111111222222
type+++++-
imageC1C2C2C2C4C7C14C14C14C28D4Q8C4○D4C7×D4C7×Q8C7×C4○D4
kernelC7×C23.63C23C7×C2.C42C2×C4×C28C14×C4⋊C4C7×C4⋊C4C23.63C23C2.C42C2×C42C2×C4⋊C4C4⋊C4C2×C28C2×C28C2×C14C2×C4C2×C4C22
# reps1412862461248228121248

Matrix representation of C7×C23.63C23 ►in GL5(𝔽29)

10000
023000
002300
00010
00001
,
10000
01000
00100
000280
000028
,
10000
028000
002800
00010
00001
,
280000
028000
002800
000280
000028
,
120000
00100
028000
0001825
0001611
,
170000
012000
001200
000122
000017
,
10000
028000
00100
000170
000017

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[12,0,0,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,18,16,0,0,0,25,11],[17,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,2,17],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,0,17] >;
 

C7×C23.63C23 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{63}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.63C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,795);
 
// by ID
 
G=gap.SmallGroup(448,795);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1576,2438,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=f^2=d,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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