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G = C7×C23.83C23  order 448 = 26·7

Direct product of C7 and C23.83C23

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

Aliases: C7×C23.83C23, (C2×C28).42Q8, (C2×C28).313D4, C22.76(D4×C14), C22.25(Q8×C14), C14.92(C22⋊Q8), C14.70(C4.4D4), C23.83(C22×C14), C14.30(C42.C2), C14.36(C42⋊2C2), C2.C42.10C14, (C22×C28).405C22, (C22×C14).464C23, C14.94(C22.D4), (C2×C4).5(C7×Q8), (C2×C4).20(C7×D4), (C14×C4⋊C4).40C2, (C2×C4⋊C4).11C14, C2.8(C7×C4.4D4), C2.11(C7×C22⋊Q8), C2.5(C7×C42.C2), (C2×C14).616(C2×D4), (C22×C4).9(C2×C14), C2.6(C7×C42⋊2C2), (C2×C14).113(C2×Q8), C22.43(C7×C4○D4), (C2×C14).224(C4○D4), (C7×C2.C42).6C2, C2.10(C7×C22.D4), SmallGroup(448,808)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C7×C23.83C23
Chief series
C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C14×C4⋊C4 — C7×C23.83C23
Lower central
C1 — C23 — C7×C23.83C23
Upper central
C1 — C22×C14 — C7×C23.83C23

Generators and relations for C7×C23.83C23
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=g2=b, f2=cb=bc, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, 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: 218 in 134 conjugacy classes, 70 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×C28, C2×C28, C22×C14, C23.83C23, C7×C4⋊C4, C22×C28, C22×C28, C7×C2.C42, C7×C2.C42, C14×C4⋊C4, C7×C23.83C23
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C7×D4, C7×Q8, C22×C14, C23.83C23, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊Q8, C7×C22.D4, C7×C4.4D4, C7×C42.C2, C7×C42⋊2C2, C7×C23.83C23

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

(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 435)(23 436)(24 437)(25 438)(26 439)(27 440)(28 441)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(43 444)(44 445)(45 446)(46 447)(47 448)(48 442)(49 443)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(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)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(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)(218 239)(219 240)(220 241)(221 242)(222 243)(223 244)(224 245)(232 246)(233 247)(234 248)(235 249)(236 250)(237 251)(238 252)(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)(274 295)(275 296)(276 297)(277 298)(278 299)(279 300)(280 301)(288 302)(289 303)(290 304)(291 305)(292 306)(293 307)(294 308)(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)(330 351)(331 352)(332 353)(333 354)(334 355)(335 356)(336 357)(344 358)(345 359)(346 360)(347 361)(348 362)(349 363)(350 364)(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)(386 407)(387 408)(388 409)(389 410)(390 411)(391 412)(392 413)(400 414)(401 415)(402 416)(403 417)(404 418)(405 419)(406 420)(421 431)(422 432)(423 433)(424 434)(425 428)(426 429)(427 430)

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

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

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

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

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×C23.83C23C7×C2.C42C14×C4⋊C4C23.83C23C2.C42C2×C4⋊C4C2×C28C2×C28C2×C14C2×C4C2×C4C22
# reps152630122210121260

Matrix representation of C7×C23.83C23 ►in GL6(𝔽29)

1600000
0160000
001000
000100
000070
000007
,
100000
010000
0028000
0002800
000010
000001
,
2800000
0280000
0028000
0002800
000010
000001
,
2800000
0280000
0028000
0002800
0000280
0000028
,
2380000
2160000
0001200
0012000
0000280
0000028
,
1590000
20140000
001000
0002800
00002321
000086
,
010000
100000
000100
0028000
000001
000010

G:=sub<GL(6,GF(29))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[23,21,0,0,0,0,8,6,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[15,20,0,0,0,0,9,14,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,23,8,0,0,0,0,21,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C7×C23.83C23 in GAP, Magma, Sage, TeX 

C_7\times C_2^3._{83}C_2^3
 
% in TeX
 
G:=Group("C7xC2^3.83C2^3");
 
// GroupNames label
 
G:=SmallGroup(448,808);
 
// by ID
 
G=gap.SmallGroup(448,808);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,1184,2438,2403,310]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c*b=b*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*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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