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G = C7×C23.81C23  order 448 = 26·7

Direct product of C7 and C23.81C23

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

Aliases: C7×C23.81C23, (C2×C28).41Q8, (C2×C28).311D4, C14.39(C4⋊Q8), C22.74(D4×C14), C22.23(Q8×C14), C14.90(C22⋊Q8), C14.141(C4⋊D4), C2.C42.9C14, C23.81(C22×C14), C14.29(C42.C2), (C22×C28).404C22, (C22×C14).462C23, C14.92(C22.D4), C2.5(C7×C4⋊Q8), (C2×C4).4(C7×Q8), (C2×C4).18(C7×D4), (C14×C4⋊C4).39C2, (C2×C4⋊C4).10C14, C2.9(C7×C22⋊Q8), C2.10(C7×C4⋊D4), C2.4(C7×C42.C2), (C2×C14).614(C2×D4), (C2×C14).111(C2×Q8), C22.41(C7×C4○D4), (C22×C4).22(C2×C14), (C2×C14).222(C4○D4), C2.8(C7×C22.D4), (C7×C2.C42).28C2, SmallGroup(448,806)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C7×C23.81C23
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C4⋊C4 — C7×C23.81C23
Lower central
C1 — C23 — C7×C23.81C23
Upper central
C1 — C22×C14 — C7×C23.81C23

Generators and relations for C7×C23.81C23
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=c, f2=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, gfg-1=df=fd, dg=gd >

Subgroups: 234 in 150 conjugacy classes, 78 normal (30 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C28, C2×C14, C2×C14, C2.C42, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C23.81C23, C7×C4⋊C4, C22×C28, C22×C28, C7×C2.C42, C7×C2.C42, C14×C4⋊C4, C14×C4⋊C4, C7×C23.81C23
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C7×D4, C7×Q8, C22×C14, C23.81C23, D4×C14, Q8×C14, C7×C4○D4, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4, C7×C42.C2, C7×C4⋊Q8, C7×C23.81C23

Smallest permutation representation of C7×C23.81C23
►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 28)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(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 83)(65 84)(66 78)(67 79)(68 80)(69 81)(70 82)(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)(120 139)(121 140)(122 134)(123 135)(124 136)(125 137)(126 138)(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)(176 195)(177 196)(178 190)(179 191)(180 192)(181 193)(182 194)(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)(232 251)(233 252)(234 246)(235 247)(236 248)(237 249)(238 250)(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)(288 307)(289 308)(290 302)(291 303)(292 304)(293 305)(294 306)(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)(344 363)(345 364)(346 358)(347 359)(348 360)(349 361)(350 362)(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)(400 419)(401 420)(402 414)(403 415)(404 416)(405 417)(406 418)

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

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

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

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

154 conjugacy classes

class 1 2A···2G4A···4N7A···7F14A···14AP28A···28CF
order12···24···47···714···1428···28
size11···14···41···11···14···4

154 irreducible representations

dim111111222222
type++++-
imageC1C2C2C7C14C14D4Q8C4○D4C7×D4C7×Q8C7×C4○D4
kernelC7×C23.81C23C7×C2.C42C14×C4⋊C4C23.81C23C2.C42C2×C4⋊C4C2×C28C2×C28C2×C14C2×C4C2×C4C22
# reps13461824446242436

Matrix representation of C7×C23.81C23 ►in GL6(𝔽29)

700000
070000
0024000
0002400
000070
000007
,
2800000
0280000
001000
000100
000010
000001
,
2800000
0280000
001000
000100
0000280
0000028
,
2800000
0280000
0028000
0002800
0000280
0000028
,
15210000
21140000
001000
000100
000086
00002321
,
21140000
1480000
00211100
0018800
0000914
00001520
,
010000
2800000
000100
001000
000001
000010

G:=sub<GL(6,GF(29))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[15,21,0,0,0,0,21,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,23,0,0,0,0,6,21],[21,14,0,0,0,0,14,8,0,0,0,0,0,0,21,18,0,0,0,0,11,8,0,0,0,0,0,0,9,15,0,0,0,0,14,20],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C7×C23.81C23 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{81}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.81C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,806);
 
// by ID
 
G=gap.SmallGroup(448,806);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,2438,2403,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=c,f^2=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,g*f*g^-1=d*f=f*d,d*g=g*d>;
 
// generators/relations
 

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