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G = C7×C23.67C23  order 448 = 26·7

Direct product of C7 and C23.67C23

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

Aliases: C7×C23.67C23, (C2×Q8)⋊3C28, (C2×C28)⋊11Q8, C2.5(Q8×C28), (Q8×C14)⋊13C4, C14.32(C4×Q8), (C2×C28).512D4, C14.37(C4⋊Q8), (C2×C42).11C14, C22.40(D4×C14), (C22×Q8).1C14, C22.15(Q8×C14), C28.75(C22⋊C4), C14.86(C22⋊Q8), C14.66(C4.4D4), C2.C42.7C14, C23.64(C22×C14), C22.40(C22×C28), (C22×C28).577C22, (C22×C14).455C23, (C2×C4)⋊3(C7×Q8), C2.3(C7×C4⋊Q8), (C2×C4×C28).34C2, (C2×C4⋊C4).8C14, (C2×C4).67(C7×D4), C4.7(C7×C22⋊C4), (C14×C4⋊C4).37C2, (Q8×C2×C14).11C2, (C2×C4).17(C2×C28), C2.5(C7×C22⋊Q8), C2.9(C14×C22⋊C4), C2.4(C7×C4.4D4), (C2×C28).266(C2×C4), (C2×C14).607(C2×D4), C14.96(C2×C22⋊C4), (C2×C14).107(C2×Q8), C22.25(C7×C4○D4), (C22×C4).91(C2×C14), (C2×C14).215(C4○D4), (C2×C14).227(C22×C4), (C7×C2.C42).26C2, SmallGroup(448,799)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C23.67C23
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C7×C2.C42 — C7×C23.67C23
Lower central
C1 — C22 — C7×C23.67C23
Upper central
C1 — C22×C14 — C7×C23.67C23

Generators and relations for C7×C23.67C23
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=d, f2=bcd, g2=c, 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: 274 in 186 conjugacy classes, 106 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C22×Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C23.67C23, C4×C28, C7×C4⋊C4, C22×C28, C22×C28, Q8×C14, Q8×C14, C7×C2.C42, C2×C4×C28, C14×C4⋊C4, Q8×C2×C14, C7×C23.67C23
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C22⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C22⋊C4, C4×Q8, C22⋊Q8, C4.4D4, C4⋊Q8, C2×C28, C7×D4, C7×Q8, C22×C14, C23.67C23, C7×C22⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C22⋊C4, Q8×C28, C7×C22⋊Q8, C7×C4.4D4, C7×C4⋊Q8, C7×C23.67C23

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

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

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

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

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

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

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

dim111111111111222222
type++++++-
imageC1C2C2C2C2C4C7C14C14C14C14C28D4Q8C4○D4C7×D4C7×Q8C7×C4○D4
kernelC7×C23.67C23C7×C2.C42C2×C4×C28C14×C4⋊C4Q8×C2×C14Q8×C14C23.67C23C2.C42C2×C42C2×C4⋊C4C22×Q8C2×Q8C2×C28C2×C28C2×C14C2×C4C2×C4C22
# reps14111862466648444242424

Matrix representation of C7×C23.67C23 ►in GL5(𝔽29)

10000
01000
00100
000240
000024
,
10000
028000
002800
00010
00001
,
10000
01000
00100
000280
000028
,
280000
028000
002800
00010
00001
,
170000
0112100
081800
000127
000028
,
170000
00100
01000
000120
000012
,
280000
01000
00100
000282
000281

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,24,0,0,0,0,0,24],[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],[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],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[17,0,0,0,0,0,11,8,0,0,0,21,18,0,0,0,0,0,1,0,0,0,0,27,28],[17,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,28,0,0,0,2,1] >;
 

C7×C23.67C23 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{67}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.67C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,799);
 
// by ID
 
G=gap.SmallGroup(448,799);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2752,2438,604]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=d,f^2=b*c*d,g^2=c,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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