direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C42⋊9C4, C42⋊9C28, C28⋊8(C4⋊C4), (C4×C28)⋊19C4, (C2×C28).72Q8, (C2×C28).412D4, C14.35(C4⋊Q8), (C2×C42).9C14, C22.33(D4×C14), C22.11(Q8×C14), C14.38(C4⋊1D4), C22.33(C22×C28), C23.57(C22×C14), (C22×C14).448C23, (C22×C28).574C22, C4⋊1(C7×C4⋊C4), C2.1(C7×C4⋊Q8), C2.6(C14×C4⋊C4), (C2×C4×C28).32C2, (C2×C4⋊C4).5C14, C14.61(C2×C4⋊C4), (C2×C4).66(C7×D4), C2.1(C7×C4⋊1D4), (C14×C4⋊C4).34C2, (C2×C4).15(C7×Q8), (C2×C4).70(C2×C28), (C2×C28).331(C2×C4), (C2×C14).600(C2×D4), (C2×C14).103(C2×Q8), (C22×C4).107(C2×C14), (C2×C14).220(C22×C4), SmallGroup(448,792)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42⋊9C4
G = < a,b,c,d | a7=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 258 in 186 conjugacy classes, 130 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C42, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C42⋊9C4, C4×C28, C7×C4⋊C4, C22×C28, C2×C4×C28, C14×C4⋊C4, C7×C42⋊9C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C4⋊C4, C4⋊1D4, C4⋊Q8, C2×C28, C7×D4, C7×Q8, C22×C14, C42⋊9C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C14×C4⋊C4, C7×C4⋊1D4, C7×C4⋊Q8, C7×C42⋊9C4
►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)
(1 162 59 181)(2 163 60 182)(3 164 61 176)(4 165 62 177)(5 166 63 178)(6 167 57 179)(7 168 58 180)(8 353 435 358)(9 354 436 359)(10 355 437 360)(11 356 438 361)(12 357 439 362)(13 351 440 363)(14 352 441 364)(15 336 26 348)(16 330 27 349)(17 331 28 350)(18 332 22 344)(19 333 23 345)(20 334 24 346)(21 335 25 347)(29 328 48 365)(30 329 49 366)(31 323 43 367)(32 324 44 368)(33 325 45 369)(34 326 46 370)(35 327 47 371)(36 343 446 372)(37 337 447 373)(38 338 448 374)(39 339 442 375)(40 340 443 376)(41 341 444 377)(42 342 445 378)(50 207 69 171)(51 208 70 172)(52 209 64 173)(53 210 65 174)(54 204 66 175)(55 205 67 169)(56 206 68 170)(71 197 83 160)(72 198 84 161)(73 199 78 155)(74 200 79 156)(75 201 80 157)(76 202 81 158)(77 203 82 159)(85 183 95 195)(86 184 96 196)(87 185 97 190)(88 186 98 191)(89 187 92 192)(90 188 93 193)(91 189 94 194)(99 246 143 241)(100 247 144 242)(101 248 145 243)(102 249 146 244)(103 250 147 245)(104 251 141 239)(105 252 142 240)(106 263 125 227)(107 264 126 228)(108 265 120 229)(109 266 121 230)(110 260 122 231)(111 261 123 225)(112 262 124 226)(113 235 149 223)(114 236 150 224)(115 237 151 218)(116 238 152 219)(117 232 153 220)(118 233 154 221)(119 234 148 222)(127 253 139 216)(128 254 140 217)(129 255 134 211)(130 256 135 212)(131 257 136 213)(132 258 137 214)(133 259 138 215)(267 409 311 414)(268 410 312 415)(269 411 313 416)(270 412 314 417)(271 413 315 418)(272 407 309 419)(273 408 310 420)(274 395 293 431)(275 396 294 432)(276 397 288 433)(277 398 289 434)(278 399 290 428)(279 393 291 429)(280 394 292 430)(281 391 317 403)(282 392 318 404)(283 386 319 405)(284 387 320 406)(285 388 321 400)(286 389 322 401)(287 390 316 402)(295 384 307 421)(296 385 308 422)(297 379 302 423)(298 380 303 424)(299 381 304 425)(300 382 305 426)(301 383 306 427)
(1 139 85 125)(2 140 86 126)(3 134 87 120)(4 135 88 121)(5 136 89 122)(6 137 90 123)(7 138 91 124)(8 400 18 414)(9 401 19 415)(10 402 20 416)(11 403 21 417)(12 404 15 418)(13 405 16 419)(14 406 17 420)(22 409 435 388)(23 410 436 389)(24 411 437 390)(25 412 438 391)(26 413 439 392)(27 407 440 386)(28 408 441 387)(29 431 39 421)(30 432 40 422)(31 433 41 423)(32 434 42 424)(33 428 36 425)(34 429 37 426)(35 430 38 427)(43 397 444 379)(44 398 445 380)(45 399 446 381)(46 393 447 382)(47 394 448 383)(48 395 442 384)(49 396 443 385)(50 104 71 115)(51 105 72 116)(52 99 73 117)(53 100 74 118)(54 101 75 119)(55 102 76 113)(56 103 77 114)(57 132 93 111)(58 133 94 112)(59 127 95 106)(60 128 96 107)(61 129 97 108)(62 130 98 109)(63 131 92 110)(64 143 78 153)(65 144 79 154)(66 145 80 148)(67 146 81 149)(68 147 82 150)(69 141 83 151)(70 142 84 152)(155 220 173 241)(156 221 174 242)(157 222 175 243)(158 223 169 244)(159 224 170 245)(160 218 171 239)(161 219 172 240)(162 216 183 227)(163 217 184 228)(164 211 185 229)(165 212 186 230)(166 213 187 231)(167 214 188 225)(168 215 189 226)(176 255 190 265)(177 256 191 266)(178 257 192 260)(179 258 193 261)(180 259 194 262)(181 253 195 263)(182 254 196 264)(197 237 207 251)(198 238 208 252)(199 232 209 246)(200 233 210 247)(201 234 204 248)(202 235 205 249)(203 236 206 250)(267 353 285 332)(268 354 286 333)(269 355 287 334)(270 356 281 335)(271 357 282 336)(272 351 283 330)(273 352 284 331)(274 339 295 328)(275 340 296 329)(276 341 297 323)(277 342 298 324)(278 343 299 325)(279 337 300 326)(280 338 301 327)(288 377 302 367)(289 378 303 368)(290 372 304 369)(291 373 305 370)(292 374 306 371)(293 375 307 365)(294 376 308 366)(309 363 319 349)(310 364 320 350)(311 358 321 344)(312 359 322 345)(313 360 316 346)(314 361 317 347)(315 362 318 348)
(1 293 69 319)(2 294 70 320)(3 288 64 321)(4 289 65 322)(5 290 66 316)(6 291 67 317)(7 292 68 318)(8 255 31 246)(9 256 32 247)(10 257 33 248)(11 258 34 249)(12 259 35 250)(13 253 29 251)(14 254 30 252)(15 262 38 236)(16 263 39 237)(17 264 40 238)(18 265 41 232)(19 266 42 233)(20 260 36 234)(21 261 37 235)(22 229 444 220)(23 230 445 221)(24 231 446 222)(25 225 447 223)(26 226 448 224)(27 227 442 218)(28 228 443 219)(43 241 435 211)(44 242 436 212)(45 243 437 213)(46 244 438 214)(47 245 439 215)(48 239 440 216)(49 240 441 217)(50 283 59 274)(51 284 60 275)(52 285 61 276)(53 286 62 277)(54 287 63 278)(55 281 57 279)(56 282 58 280)(71 272 95 295)(72 273 96 296)(73 267 97 297)(74 268 98 298)(75 269 92 299)(76 270 93 300)(77 271 94 301)(78 311 87 302)(79 312 88 303)(80 313 89 304)(81 314 90 305)(82 315 91 306)(83 309 85 307)(84 310 86 308)(99 353 129 323)(100 354 130 324)(101 355 131 325)(102 356 132 326)(103 357 133 327)(104 351 127 328)(105 352 128 329)(106 339 115 330)(107 340 116 331)(108 341 117 332)(109 342 118 333)(110 343 119 334)(111 337 113 335)(112 338 114 336)(120 377 153 344)(121 378 154 345)(122 372 148 346)(123 373 149 347)(124 374 150 348)(125 375 151 349)(126 376 152 350)(134 367 143 358)(135 368 144 359)(136 369 145 360)(137 370 146 361)(138 371 147 362)(139 365 141 363)(140 366 142 364)(155 409 185 379)(156 410 186 380)(157 411 187 381)(158 412 188 382)(159 413 189 383)(160 407 183 384)(161 408 184 385)(162 395 171 386)(163 396 172 387)(164 397 173 388)(165 398 174 389)(166 399 175 390)(167 393 169 391)(168 394 170 392)(176 433 209 400)(177 434 210 401)(178 428 204 402)(179 429 205 403)(180 430 206 404)(181 431 207 405)(182 432 208 406)(190 423 199 414)(191 424 200 415)(192 425 201 416)(193 426 202 417)(194 427 203 418)(195 421 197 419)(196 422 198 420)
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), (1,162,59,181)(2,163,60,182)(3,164,61,176)(4,165,62,177)(5,166,63,178)(6,167,57,179)(7,168,58,180)(8,353,435,358)(9,354,436,359)(10,355,437,360)(11,356,438,361)(12,357,439,362)(13,351,440,363)(14,352,441,364)(15,336,26,348)(16,330,27,349)(17,331,28,350)(18,332,22,344)(19,333,23,345)(20,334,24,346)(21,335,25,347)(29,328,48,365)(30,329,49,366)(31,323,43,367)(32,324,44,368)(33,325,45,369)(34,326,46,370)(35,327,47,371)(36,343,446,372)(37,337,447,373)(38,338,448,374)(39,339,442,375)(40,340,443,376)(41,341,444,377)(42,342,445,378)(50,207,69,171)(51,208,70,172)(52,209,64,173)(53,210,65,174)(54,204,66,175)(55,205,67,169)(56,206,68,170)(71,197,83,160)(72,198,84,161)(73,199,78,155)(74,200,79,156)(75,201,80,157)(76,202,81,158)(77,203,82,159)(85,183,95,195)(86,184,96,196)(87,185,97,190)(88,186,98,191)(89,187,92,192)(90,188,93,193)(91,189,94,194)(99,246,143,241)(100,247,144,242)(101,248,145,243)(102,249,146,244)(103,250,147,245)(104,251,141,239)(105,252,142,240)(106,263,125,227)(107,264,126,228)(108,265,120,229)(109,266,121,230)(110,260,122,231)(111,261,123,225)(112,262,124,226)(113,235,149,223)(114,236,150,224)(115,237,151,218)(116,238,152,219)(117,232,153,220)(118,233,154,221)(119,234,148,222)(127,253,139,216)(128,254,140,217)(129,255,134,211)(130,256,135,212)(131,257,136,213)(132,258,137,214)(133,259,138,215)(267,409,311,414)(268,410,312,415)(269,411,313,416)(270,412,314,417)(271,413,315,418)(272,407,309,419)(273,408,310,420)(274,395,293,431)(275,396,294,432)(276,397,288,433)(277,398,289,434)(278,399,290,428)(279,393,291,429)(280,394,292,430)(281,391,317,403)(282,392,318,404)(283,386,319,405)(284,387,320,406)(285,388,321,400)(286,389,322,401)(287,390,316,402)(295,384,307,421)(296,385,308,422)(297,379,302,423)(298,380,303,424)(299,381,304,425)(300,382,305,426)(301,383,306,427), (1,139,85,125)(2,140,86,126)(3,134,87,120)(4,135,88,121)(5,136,89,122)(6,137,90,123)(7,138,91,124)(8,400,18,414)(9,401,19,415)(10,402,20,416)(11,403,21,417)(12,404,15,418)(13,405,16,419)(14,406,17,420)(22,409,435,388)(23,410,436,389)(24,411,437,390)(25,412,438,391)(26,413,439,392)(27,407,440,386)(28,408,441,387)(29,431,39,421)(30,432,40,422)(31,433,41,423)(32,434,42,424)(33,428,36,425)(34,429,37,426)(35,430,38,427)(43,397,444,379)(44,398,445,380)(45,399,446,381)(46,393,447,382)(47,394,448,383)(48,395,442,384)(49,396,443,385)(50,104,71,115)(51,105,72,116)(52,99,73,117)(53,100,74,118)(54,101,75,119)(55,102,76,113)(56,103,77,114)(57,132,93,111)(58,133,94,112)(59,127,95,106)(60,128,96,107)(61,129,97,108)(62,130,98,109)(63,131,92,110)(64,143,78,153)(65,144,79,154)(66,145,80,148)(67,146,81,149)(68,147,82,150)(69,141,83,151)(70,142,84,152)(155,220,173,241)(156,221,174,242)(157,222,175,243)(158,223,169,244)(159,224,170,245)(160,218,171,239)(161,219,172,240)(162,216,183,227)(163,217,184,228)(164,211,185,229)(165,212,186,230)(166,213,187,231)(167,214,188,225)(168,215,189,226)(176,255,190,265)(177,256,191,266)(178,257,192,260)(179,258,193,261)(180,259,194,262)(181,253,195,263)(182,254,196,264)(197,237,207,251)(198,238,208,252)(199,232,209,246)(200,233,210,247)(201,234,204,248)(202,235,205,249)(203,236,206,250)(267,353,285,332)(268,354,286,333)(269,355,287,334)(270,356,281,335)(271,357,282,336)(272,351,283,330)(273,352,284,331)(274,339,295,328)(275,340,296,329)(276,341,297,323)(277,342,298,324)(278,343,299,325)(279,337,300,326)(280,338,301,327)(288,377,302,367)(289,378,303,368)(290,372,304,369)(291,373,305,370)(292,374,306,371)(293,375,307,365)(294,376,308,366)(309,363,319,349)(310,364,320,350)(311,358,321,344)(312,359,322,345)(313,360,316,346)(314,361,317,347)(315,362,318,348), (1,293,69,319)(2,294,70,320)(3,288,64,321)(4,289,65,322)(5,290,66,316)(6,291,67,317)(7,292,68,318)(8,255,31,246)(9,256,32,247)(10,257,33,248)(11,258,34,249)(12,259,35,250)(13,253,29,251)(14,254,30,252)(15,262,38,236)(16,263,39,237)(17,264,40,238)(18,265,41,232)(19,266,42,233)(20,260,36,234)(21,261,37,235)(22,229,444,220)(23,230,445,221)(24,231,446,222)(25,225,447,223)(26,226,448,224)(27,227,442,218)(28,228,443,219)(43,241,435,211)(44,242,436,212)(45,243,437,213)(46,244,438,214)(47,245,439,215)(48,239,440,216)(49,240,441,217)(50,283,59,274)(51,284,60,275)(52,285,61,276)(53,286,62,277)(54,287,63,278)(55,281,57,279)(56,282,58,280)(71,272,95,295)(72,273,96,296)(73,267,97,297)(74,268,98,298)(75,269,92,299)(76,270,93,300)(77,271,94,301)(78,311,87,302)(79,312,88,303)(80,313,89,304)(81,314,90,305)(82,315,91,306)(83,309,85,307)(84,310,86,308)(99,353,129,323)(100,354,130,324)(101,355,131,325)(102,356,132,326)(103,357,133,327)(104,351,127,328)(105,352,128,329)(106,339,115,330)(107,340,116,331)(108,341,117,332)(109,342,118,333)(110,343,119,334)(111,337,113,335)(112,338,114,336)(120,377,153,344)(121,378,154,345)(122,372,148,346)(123,373,149,347)(124,374,150,348)(125,375,151,349)(126,376,152,350)(134,367,143,358)(135,368,144,359)(136,369,145,360)(137,370,146,361)(138,371,147,362)(139,365,141,363)(140,366,142,364)(155,409,185,379)(156,410,186,380)(157,411,187,381)(158,412,188,382)(159,413,189,383)(160,407,183,384)(161,408,184,385)(162,395,171,386)(163,396,172,387)(164,397,173,388)(165,398,174,389)(166,399,175,390)(167,393,169,391)(168,394,170,392)(176,433,209,400)(177,434,210,401)(178,428,204,402)(179,429,205,403)(180,430,206,404)(181,431,207,405)(182,432,208,406)(190,423,199,414)(191,424,200,415)(192,425,201,416)(193,426,202,417)(194,427,203,418)(195,421,197,419)(196,422,198,420)>;
G:=Group( (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), (1,162,59,181)(2,163,60,182)(3,164,61,176)(4,165,62,177)(5,166,63,178)(6,167,57,179)(7,168,58,180)(8,353,435,358)(9,354,436,359)(10,355,437,360)(11,356,438,361)(12,357,439,362)(13,351,440,363)(14,352,441,364)(15,336,26,348)(16,330,27,349)(17,331,28,350)(18,332,22,344)(19,333,23,345)(20,334,24,346)(21,335,25,347)(29,328,48,365)(30,329,49,366)(31,323,43,367)(32,324,44,368)(33,325,45,369)(34,326,46,370)(35,327,47,371)(36,343,446,372)(37,337,447,373)(38,338,448,374)(39,339,442,375)(40,340,443,376)(41,341,444,377)(42,342,445,378)(50,207,69,171)(51,208,70,172)(52,209,64,173)(53,210,65,174)(54,204,66,175)(55,205,67,169)(56,206,68,170)(71,197,83,160)(72,198,84,161)(73,199,78,155)(74,200,79,156)(75,201,80,157)(76,202,81,158)(77,203,82,159)(85,183,95,195)(86,184,96,196)(87,185,97,190)(88,186,98,191)(89,187,92,192)(90,188,93,193)(91,189,94,194)(99,246,143,241)(100,247,144,242)(101,248,145,243)(102,249,146,244)(103,250,147,245)(104,251,141,239)(105,252,142,240)(106,263,125,227)(107,264,126,228)(108,265,120,229)(109,266,121,230)(110,260,122,231)(111,261,123,225)(112,262,124,226)(113,235,149,223)(114,236,150,224)(115,237,151,218)(116,238,152,219)(117,232,153,220)(118,233,154,221)(119,234,148,222)(127,253,139,216)(128,254,140,217)(129,255,134,211)(130,256,135,212)(131,257,136,213)(132,258,137,214)(133,259,138,215)(267,409,311,414)(268,410,312,415)(269,411,313,416)(270,412,314,417)(271,413,315,418)(272,407,309,419)(273,408,310,420)(274,395,293,431)(275,396,294,432)(276,397,288,433)(277,398,289,434)(278,399,290,428)(279,393,291,429)(280,394,292,430)(281,391,317,403)(282,392,318,404)(283,386,319,405)(284,387,320,406)(285,388,321,400)(286,389,322,401)(287,390,316,402)(295,384,307,421)(296,385,308,422)(297,379,302,423)(298,380,303,424)(299,381,304,425)(300,382,305,426)(301,383,306,427), (1,139,85,125)(2,140,86,126)(3,134,87,120)(4,135,88,121)(5,136,89,122)(6,137,90,123)(7,138,91,124)(8,400,18,414)(9,401,19,415)(10,402,20,416)(11,403,21,417)(12,404,15,418)(13,405,16,419)(14,406,17,420)(22,409,435,388)(23,410,436,389)(24,411,437,390)(25,412,438,391)(26,413,439,392)(27,407,440,386)(28,408,441,387)(29,431,39,421)(30,432,40,422)(31,433,41,423)(32,434,42,424)(33,428,36,425)(34,429,37,426)(35,430,38,427)(43,397,444,379)(44,398,445,380)(45,399,446,381)(46,393,447,382)(47,394,448,383)(48,395,442,384)(49,396,443,385)(50,104,71,115)(51,105,72,116)(52,99,73,117)(53,100,74,118)(54,101,75,119)(55,102,76,113)(56,103,77,114)(57,132,93,111)(58,133,94,112)(59,127,95,106)(60,128,96,107)(61,129,97,108)(62,130,98,109)(63,131,92,110)(64,143,78,153)(65,144,79,154)(66,145,80,148)(67,146,81,149)(68,147,82,150)(69,141,83,151)(70,142,84,152)(155,220,173,241)(156,221,174,242)(157,222,175,243)(158,223,169,244)(159,224,170,245)(160,218,171,239)(161,219,172,240)(162,216,183,227)(163,217,184,228)(164,211,185,229)(165,212,186,230)(166,213,187,231)(167,214,188,225)(168,215,189,226)(176,255,190,265)(177,256,191,266)(178,257,192,260)(179,258,193,261)(180,259,194,262)(181,253,195,263)(182,254,196,264)(197,237,207,251)(198,238,208,252)(199,232,209,246)(200,233,210,247)(201,234,204,248)(202,235,205,249)(203,236,206,250)(267,353,285,332)(268,354,286,333)(269,355,287,334)(270,356,281,335)(271,357,282,336)(272,351,283,330)(273,352,284,331)(274,339,295,328)(275,340,296,329)(276,341,297,323)(277,342,298,324)(278,343,299,325)(279,337,300,326)(280,338,301,327)(288,377,302,367)(289,378,303,368)(290,372,304,369)(291,373,305,370)(292,374,306,371)(293,375,307,365)(294,376,308,366)(309,363,319,349)(310,364,320,350)(311,358,321,344)(312,359,322,345)(313,360,316,346)(314,361,317,347)(315,362,318,348), (1,293,69,319)(2,294,70,320)(3,288,64,321)(4,289,65,322)(5,290,66,316)(6,291,67,317)(7,292,68,318)(8,255,31,246)(9,256,32,247)(10,257,33,248)(11,258,34,249)(12,259,35,250)(13,253,29,251)(14,254,30,252)(15,262,38,236)(16,263,39,237)(17,264,40,238)(18,265,41,232)(19,266,42,233)(20,260,36,234)(21,261,37,235)(22,229,444,220)(23,230,445,221)(24,231,446,222)(25,225,447,223)(26,226,448,224)(27,227,442,218)(28,228,443,219)(43,241,435,211)(44,242,436,212)(45,243,437,213)(46,244,438,214)(47,245,439,215)(48,239,440,216)(49,240,441,217)(50,283,59,274)(51,284,60,275)(52,285,61,276)(53,286,62,277)(54,287,63,278)(55,281,57,279)(56,282,58,280)(71,272,95,295)(72,273,96,296)(73,267,97,297)(74,268,98,298)(75,269,92,299)(76,270,93,300)(77,271,94,301)(78,311,87,302)(79,312,88,303)(80,313,89,304)(81,314,90,305)(82,315,91,306)(83,309,85,307)(84,310,86,308)(99,353,129,323)(100,354,130,324)(101,355,131,325)(102,356,132,326)(103,357,133,327)(104,351,127,328)(105,352,128,329)(106,339,115,330)(107,340,116,331)(108,341,117,332)(109,342,118,333)(110,343,119,334)(111,337,113,335)(112,338,114,336)(120,377,153,344)(121,378,154,345)(122,372,148,346)(123,373,149,347)(124,374,150,348)(125,375,151,349)(126,376,152,350)(134,367,143,358)(135,368,144,359)(136,369,145,360)(137,370,146,361)(138,371,147,362)(139,365,141,363)(140,366,142,364)(155,409,185,379)(156,410,186,380)(157,411,187,381)(158,412,188,382)(159,413,189,383)(160,407,183,384)(161,408,184,385)(162,395,171,386)(163,396,172,387)(164,397,173,388)(165,398,174,389)(166,399,175,390)(167,393,169,391)(168,394,170,392)(176,433,209,400)(177,434,210,401)(178,428,204,402)(179,429,205,403)(180,430,206,404)(181,431,207,405)(182,432,208,406)(190,423,199,414)(191,424,200,415)(192,425,201,416)(193,426,202,417)(194,427,203,418)(195,421,197,419)(196,422,198,420) );
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)], [(1,162,59,181),(2,163,60,182),(3,164,61,176),(4,165,62,177),(5,166,63,178),(6,167,57,179),(7,168,58,180),(8,353,435,358),(9,354,436,359),(10,355,437,360),(11,356,438,361),(12,357,439,362),(13,351,440,363),(14,352,441,364),(15,336,26,348),(16,330,27,349),(17,331,28,350),(18,332,22,344),(19,333,23,345),(20,334,24,346),(21,335,25,347),(29,328,48,365),(30,329,49,366),(31,323,43,367),(32,324,44,368),(33,325,45,369),(34,326,46,370),(35,327,47,371),(36,343,446,372),(37,337,447,373),(38,338,448,374),(39,339,442,375),(40,340,443,376),(41,341,444,377),(42,342,445,378),(50,207,69,171),(51,208,70,172),(52,209,64,173),(53,210,65,174),(54,204,66,175),(55,205,67,169),(56,206,68,170),(71,197,83,160),(72,198,84,161),(73,199,78,155),(74,200,79,156),(75,201,80,157),(76,202,81,158),(77,203,82,159),(85,183,95,195),(86,184,96,196),(87,185,97,190),(88,186,98,191),(89,187,92,192),(90,188,93,193),(91,189,94,194),(99,246,143,241),(100,247,144,242),(101,248,145,243),(102,249,146,244),(103,250,147,245),(104,251,141,239),(105,252,142,240),(106,263,125,227),(107,264,126,228),(108,265,120,229),(109,266,121,230),(110,260,122,231),(111,261,123,225),(112,262,124,226),(113,235,149,223),(114,236,150,224),(115,237,151,218),(116,238,152,219),(117,232,153,220),(118,233,154,221),(119,234,148,222),(127,253,139,216),(128,254,140,217),(129,255,134,211),(130,256,135,212),(131,257,136,213),(132,258,137,214),(133,259,138,215),(267,409,311,414),(268,410,312,415),(269,411,313,416),(270,412,314,417),(271,413,315,418),(272,407,309,419),(273,408,310,420),(274,395,293,431),(275,396,294,432),(276,397,288,433),(277,398,289,434),(278,399,290,428),(279,393,291,429),(280,394,292,430),(281,391,317,403),(282,392,318,404),(283,386,319,405),(284,387,320,406),(285,388,321,400),(286,389,322,401),(287,390,316,402),(295,384,307,421),(296,385,308,422),(297,379,302,423),(298,380,303,424),(299,381,304,425),(300,382,305,426),(301,383,306,427)], [(1,139,85,125),(2,140,86,126),(3,134,87,120),(4,135,88,121),(5,136,89,122),(6,137,90,123),(7,138,91,124),(8,400,18,414),(9,401,19,415),(10,402,20,416),(11,403,21,417),(12,404,15,418),(13,405,16,419),(14,406,17,420),(22,409,435,388),(23,410,436,389),(24,411,437,390),(25,412,438,391),(26,413,439,392),(27,407,440,386),(28,408,441,387),(29,431,39,421),(30,432,40,422),(31,433,41,423),(32,434,42,424),(33,428,36,425),(34,429,37,426),(35,430,38,427),(43,397,444,379),(44,398,445,380),(45,399,446,381),(46,393,447,382),(47,394,448,383),(48,395,442,384),(49,396,443,385),(50,104,71,115),(51,105,72,116),(52,99,73,117),(53,100,74,118),(54,101,75,119),(55,102,76,113),(56,103,77,114),(57,132,93,111),(58,133,94,112),(59,127,95,106),(60,128,96,107),(61,129,97,108),(62,130,98,109),(63,131,92,110),(64,143,78,153),(65,144,79,154),(66,145,80,148),(67,146,81,149),(68,147,82,150),(69,141,83,151),(70,142,84,152),(155,220,173,241),(156,221,174,242),(157,222,175,243),(158,223,169,244),(159,224,170,245),(160,218,171,239),(161,219,172,240),(162,216,183,227),(163,217,184,228),(164,211,185,229),(165,212,186,230),(166,213,187,231),(167,214,188,225),(168,215,189,226),(176,255,190,265),(177,256,191,266),(178,257,192,260),(179,258,193,261),(180,259,194,262),(181,253,195,263),(182,254,196,264),(197,237,207,251),(198,238,208,252),(199,232,209,246),(200,233,210,247),(201,234,204,248),(202,235,205,249),(203,236,206,250),(267,353,285,332),(268,354,286,333),(269,355,287,334),(270,356,281,335),(271,357,282,336),(272,351,283,330),(273,352,284,331),(274,339,295,328),(275,340,296,329),(276,341,297,323),(277,342,298,324),(278,343,299,325),(279,337,300,326),(280,338,301,327),(288,377,302,367),(289,378,303,368),(290,372,304,369),(291,373,305,370),(292,374,306,371),(293,375,307,365),(294,376,308,366),(309,363,319,349),(310,364,320,350),(311,358,321,344),(312,359,322,345),(313,360,316,346),(314,361,317,347),(315,362,318,348)], [(1,293,69,319),(2,294,70,320),(3,288,64,321),(4,289,65,322),(5,290,66,316),(6,291,67,317),(7,292,68,318),(8,255,31,246),(9,256,32,247),(10,257,33,248),(11,258,34,249),(12,259,35,250),(13,253,29,251),(14,254,30,252),(15,262,38,236),(16,263,39,237),(17,264,40,238),(18,265,41,232),(19,266,42,233),(20,260,36,234),(21,261,37,235),(22,229,444,220),(23,230,445,221),(24,231,446,222),(25,225,447,223),(26,226,448,224),(27,227,442,218),(28,228,443,219),(43,241,435,211),(44,242,436,212),(45,243,437,213),(46,244,438,214),(47,245,439,215),(48,239,440,216),(49,240,441,217),(50,283,59,274),(51,284,60,275),(52,285,61,276),(53,286,62,277),(54,287,63,278),(55,281,57,279),(56,282,58,280),(71,272,95,295),(72,273,96,296),(73,267,97,297),(74,268,98,298),(75,269,92,299),(76,270,93,300),(77,271,94,301),(78,311,87,302),(79,312,88,303),(80,313,89,304),(81,314,90,305),(82,315,91,306),(83,309,85,307),(84,310,86,308),(99,353,129,323),(100,354,130,324),(101,355,131,325),(102,356,132,326),(103,357,133,327),(104,351,127,328),(105,352,128,329),(106,339,115,330),(107,340,116,331),(108,341,117,332),(109,342,118,333),(110,343,119,334),(111,337,113,335),(112,338,114,336),(120,377,153,344),(121,378,154,345),(122,372,148,346),(123,373,149,347),(124,374,150,348),(125,375,151,349),(126,376,152,350),(134,367,143,358),(135,368,144,359),(136,369,145,360),(137,370,146,361),(138,371,147,362),(139,365,141,363),(140,366,142,364),(155,409,185,379),(156,410,186,380),(157,411,187,381),(158,412,188,382),(159,413,189,383),(160,407,183,384),(161,408,184,385),(162,395,171,386),(163,396,172,387),(164,397,173,388),(165,398,174,389),(166,399,175,390),(167,393,169,391),(168,394,170,392),(176,433,209,400),(177,434,210,401),(178,428,204,402),(179,429,205,403),(180,430,206,404),(181,431,207,405),(182,432,208,406),(190,423,199,414),(191,424,200,415),(192,425,201,416),(193,426,202,417),(194,427,203,418),(195,421,197,419),(196,422,198,420)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 4M | ··· | 4T | 7A | ··· | 7F | 14A | ··· | 14AP | 28A | ··· | 28BT | 28BU | ··· | 28DP |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | Q8 | C7×D4 | C7×Q8 |
| kernel | C7×C42⋊9C4 | C2×C4×C28 | C14×C4⋊C4 | C4×C28 | C42⋊9C4 | C2×C42 | C2×C4⋊C4 | C42 | C2×C28 | C2×C28 | C2×C4 | C2×C4 |
| # reps | 1 | 1 | 6 | 8 | 6 | 6 | 36 | 48 | 6 | 6 | 36 | 36 |
Matrix representation of C7×C42⋊9C4 ►in GL6(𝔽29)
| 20 | 0 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 0 | 25 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 27 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 |
| 0 | 0 | 10 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 15 |
| 0 | 0 | 0 | 0 | 19 | 27 |
G:=sub<GL(6,GF(29))| [20,0,0,0,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,0,0,0,0,0,0,25],[17,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,27,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,2,17,0,0,0,0,0,0,17,0,0,0,0,0,1,12],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,4,10,0,0,0,0,13,25,0,0,0,0,0,0,2,19,0,0,0,0,15,27] >;
C7×C42⋊9C4 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes_9C_4
% in TeX
G:=Group("C7xC4^2:9C4"); // GroupNames label
G:=SmallGroup(448,792);
// by ID
G=gap.SmallGroup(448,792);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,400,2438,604]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations