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G = C7×C23.65C23  order 448 = 26·7

Direct product of C7 and C23.65C23

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

Aliases: C7×C23.65C23, C4⋊C4⋊6C28, C28⋊9(C4⋊C4), C2.8(D4×C28), C2.4(Q8×C28), C14.31(C4×Q8), (C2×C28).73Q8, C14.109(C4×D4), (C2×C28).359D4, C14.36(C4⋊Q8), (C2×C42).10C14, C22.38(D4×C14), C22.14(Q8×C14), C14.85(C22⋊Q8), C14.135(C4⋊D4), C2.C42.6C14, C22.38(C22×C28), C23.62(C22×C14), C14.28(C42.C2), (C22×C14).453C23, (C22×C28).575C22, C4⋊2(C7×C4⋊C4), (C7×C4⋊C4)⋊13C4, C2.2(C7×C4⋊Q8), C2.8(C14×C4⋊C4), (C2×C4×C28).33C2, (C2×C4⋊C4).7C14, C14.63(C2×C4⋊C4), C2.4(C7×C4⋊D4), (C14×C4⋊C4).36C2, (C2×C4).11(C7×Q8), (C2×C4).16(C2×C28), C2.4(C7×C22⋊Q8), (C2×C4).117(C7×D4), (C2×C28).264(C2×C4), C2.3(C7×C42.C2), (C2×C14).605(C2×D4), (C2×C14).106(C2×Q8), C22.23(C7×C4○D4), (C22×C4).89(C2×C14), (C2×C14).213(C4○D4), (C2×C14).225(C22×C4), (C7×C2.C42).25C2, SmallGroup(448,797)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C23.65C23
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C4⋊C4 — C7×C23.65C23
Lower central
C1 — C22 — C7×C23.65C23
Upper central
C1 — C22×C14 — C7×C23.65C23

Generators and relations for C7×C23.65C23
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=f2=d, 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: 242 in 170 conjugacy classes, 106 normal (46 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C28, C28, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C23.65C23, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C7×C2.C42, C2×C4×C28, C14×C4⋊C4, C14×C4⋊C4, C7×C23.65C23
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C28, C2×C14, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×C28, C7×D4, C7×Q8, C22×C14, C23.65C23, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C14×C4⋊C4, D4×C28, Q8×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C42.C2, C7×C4⋊Q8, C7×C23.65C23

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

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

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

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

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

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

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×C23.65C23C7×C2.C42C2×C4×C28C14×C4⋊C4C7×C4⋊C4C23.65C23C2.C42C2×C42C2×C4⋊C4C4⋊C4C2×C28C2×C28C2×C14C2×C4C2×C4C22
# reps1214861262448444242424

Matrix representation of C7×C23.65C23 ►in GL5(𝔽29)

10000
024000
002400
000200
000020
,
10000
028000
002800
00010
00001
,
10000
01000
00100
000280
000028
,
280000
028000
002800
000280
000028
,
170000
013200
021600
000148
000815
,
170000
00100
028000
000120
000012
,
280000
028000
002800
00001
000280

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,24,0,0,0,0,0,24,0,0,0,0,0,20,0,0,0,0,0,20],[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],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[17,0,0,0,0,0,13,2,0,0,0,2,16,0,0,0,0,0,14,8,0,0,0,8,15],[17,0,0,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,1,0] >;
 

C7×C23.65C23 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{65}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.65C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,797);
 
// by ID
 
G=gap.SmallGroup(448,797);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1576,2438,604]);
 
// 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=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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