C7xC2^3.78C2^3 - GroupNames

G = C₇×C₂³.₇₈C₂³ order 448 = 2⁶·7

Direct product of C₇ and C₂³.₇₈C₂³

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

Aliases: C₇×C₂³.₇₈C₂³, (C₂×C₂₈)⋊₈Q₈, (C₂×C₂₈).₃₀₈D₄, C₁₄.₃₈(C₄⋊Q₈), C₁₄.₉₃C₂²≀C₂, C₂².₇₁(D₄×C₁₄), (C₂²×Q₈).₂C₁₄, C₂².₂₁(Q₈×C₁₄), C₁₄.₈₈(C₂²⋊Q₈), C₂.C₄².₈C₁₄, C₂³.₇₈(C₂²×C₁₄), (C₂²×C₂₈).₄₀₃C₂², (C₂²×C₁₄).₄₅₉C₂³, (C₂×C₄)⋊₁(C₇×Q₈), C₂.₄(C₇×C₄⋊Q₈), (C₂×C₄⋊C₄).₉C₁₄, (C₂×C₄).₁₅(C₇×D₄), (C₁₄×C₄⋊C₄).₃₈C₂, (Q₈×C₂×C₁₄).₁₂C₂, C₂.₇(C₇×C₂²⋊Q₈), C₂.₇(C₇×C₂²≀C₂), (C₂×C₁₄).₆₁₁(C₂×D₄), (C₂×C₁₄).₁₀₉(C₂×Q₈), C₂².₃₈(C₇×C₄○D₄), (C₂²×C₄).₂₁(C₂×C₁₄), (C₂×C₁₄).₂₁₉(C₄○D₄), (C₇×C₂.C₄²).₂₇C₂, SmallGroup(448,803)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₇×C₂³.₇₈C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₁₄ — C₂²×C₂₈ — Q₈×C₂×C₁₄ — C₇×C₂³.₇₈C₂³

Lower central

C₁ — C₂³ — C₇×C₂³.₇₈C₂³

Upper central

C₁ — C₂²×C₁₄ — C₇×C₂³.₇₈C₂³

Generators and relations for C₇×C₂³.₇₈C₂³
G = < a,b,c,d,e,f,g | a⁷=b²=c²=d²=1, e²=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: 282 in 182 conjugacy classes, 86 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂.C₄², C₂×C₄⋊C₄, C₂²×Q₈, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×C₁₄, C₂³.₇₈C₂³, C₇×C₄⋊C₄, C₂²×C₂₈, C₂²×C₂₈, Q₈×C₁₄, C₇×C₂.C₄², C₁₄×C₄⋊C₄, Q₈×C₂×C₁₄, C₇×C₂³.₇₈C₂³
Quotients: C₁, C₂, C₂², C₇, D₄, Q₈, C₂³, C₁₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₁₄, C₂²≀C₂, C₂²⋊Q₈, C₄⋊Q₈, C₇×D₄, C₇×Q₈, C₂²×C₁₄, C₂³.₇₈C₂³, D₄×C₁₄, Q₈×C₁₄, C₇×C₄○D₄, C₇×C₂²≀C₂, C₇×C₂²⋊Q₈, C₇×C₄⋊Q₈, C₇×C₂³.₇₈C₂³

Smallest permutation representation of C₇×C₂³.₇₈C₂³
►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)

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

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

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

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

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

154 conjugacy classes

class	1	2A	···	2G	4A	···	4N	7A	···	7F	14A	···	14AP	28A	···	28CF
order	1	2	···	2	4	···	4	7	···	7	14	···	14	28	···	28
size	1	1	···	1	4	···	4	1	···	1	1	···	1	4	···	4

154 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+	+					+	-
image	C₁	C₂	C₂	C₂	C₇	C₁₄	C₁₄	C₁₄	D₄	Q₈	C₄○D₄	C₇×D₄	C₇×Q₈	C₇×C₄○D₄
kernel	C₇×C₂³.₇₈C₂³	C₇×C₂.C₄²	C₁₄×C₄⋊C₄	Q₈×C₂×C₁₄	C₂³.₇₈C₂³	C₂.C₄²	C₂×C₄⋊C₄	C₂²×Q₈	C₂×C₂₈	C₂×C₂₈	C₂×C₁₄	C₂×C₄	C₂×C₄	C₂²
# reps	1	3	3	1	6	18	18	6	6	6	2	36	36	12

Matrix representation of C₇×C₂³.₇₈C₂³ ►in GL₆(𝔽₂₉)

7	0	0	0	0	0
0	7	0	0	0	0
0	0	20	0	0	0
0	0	0	20	0	0
0	0	0	0	20	0
0	0	0	0	0	20

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

1	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	28	0
0	0	0	0	0	28

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	24	3	0	0
0	0	1	5	0	0
0	0	0	0	15	14
0	0	0	0	13	14

20	15	0	0	0	0
14	9	0	0	0	0
0	0	5	2	0	0
0	0	16	24	0	0
0	0	0	0	28	0
0	0	0	0	0	28

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	0	0	0
0	0	0	12	0	0
0	0	0	0	12	28
0	0	0	0	27	17

G:=sub<GL(6,GF(29))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,1,0,0,0,0,3,5,0,0,0,0,0,0,15,13,0,0,0,0,14,14],[20,14,0,0,0,0,15,9,0,0,0,0,0,0,5,16,0,0,0,0,2,24,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,27,0,0,0,0,28,17] >;

C₇×C₂³.₇₈C₂³ in GAP, Magma, Sage, TeX

C_7\times C_2^3._{78}C_2^3

% in TeX

G:=Group("C7xC2^3.78C2^3");

// GroupNames label

G:=SmallGroup(448,803);

// by ID

G=gap.SmallGroup(448,803);

# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,400,2438,2403,310]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=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

G = C7×C23.78C23 order 448 = 26·7

Direct product of C7 and C23.78C23

G = C₇×C₂³.₇₈C₂³ order 448 = 2⁶·7

Direct product of C₇ and C₂³.₇₈C₂³