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## G = C7×C4.SD16order 448 = 26·7

### Direct product of C7 and C4.SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×C4.SD16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — Q8×C14 — C7×Q8⋊C4 — C7×C4.SD16
 Lower central C1 — C2 — C2×C4 — C7×C4.SD16
 Upper central C1 — C2×C14 — C4×C28 — C7×C4.SD16

Generators and relations for C7×C4.SD16
G = < a,b,c,d | a7=b4=c8=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c3 >

Subgroups: 162 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C28, C28, C2×C14, C4×C8, Q8⋊C4, C4⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C4.SD16, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, Q8×C14, C4×C56, C7×Q8⋊C4, C7×C4⋊Q8, C7×C4.SD16
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, Q16, C2×D4, C4○D4, C2×C14, C4.4D4, C2×SD16, C2×Q16, C7×D4, C22×C14, C4.SD16, C7×SD16, C7×Q16, D4×C14, C7×C4○D4, C7×C4.4D4, C14×SD16, C14×Q16, C7×C4.SD16

Smallest permutation representation of C7×C4.SD16
Regular action on 448 points
Generators in S448
(1 195 87 187 79 179 71)(2 196 88 188 80 180 72)(3 197 81 189 73 181 65)(4 198 82 190 74 182 66)(5 199 83 191 75 183 67)(6 200 84 192 76 184 68)(7 193 85 185 77 177 69)(8 194 86 186 78 178 70)(9 33 141 25 133 17 125)(10 34 142 26 134 18 126)(11 35 143 27 135 19 127)(12 36 144 28 136 20 128)(13 37 137 29 129 21 121)(14 38 138 30 130 22 122)(15 39 139 31 131 23 123)(16 40 140 32 132 24 124)(41 149 175 57 165 49 157)(42 150 176 58 166 50 158)(43 151 169 59 167 51 159)(44 152 170 60 168 52 160)(45 145 171 61 161 53 153)(46 146 172 62 162 54 154)(47 147 173 63 163 55 155)(48 148 174 64 164 56 156)(89 229 221 105 213 97 205)(90 230 222 106 214 98 206)(91 231 223 107 215 99 207)(92 232 224 108 216 100 208)(93 225 217 109 209 101 201)(94 226 218 110 210 102 202)(95 227 219 111 211 103 203)(96 228 220 112 212 104 204)(113 442 330 434 322 426 314)(114 443 331 435 323 427 315)(115 444 332 436 324 428 316)(116 445 333 437 325 429 317)(117 446 334 438 326 430 318)(118 447 335 439 327 431 319)(119 448 336 440 328 432 320)(120 441 329 433 321 425 313)(233 342 282 386 274 378 266)(234 343 283 387 275 379 267)(235 344 284 388 276 380 268)(236 337 285 389 277 381 269)(237 338 286 390 278 382 270)(238 339 287 391 279 383 271)(239 340 288 392 280 384 272)(240 341 281 385 273 377 265)(241 345 369 257 361 249 353)(242 346 370 258 362 250 354)(243 347 371 259 363 251 355)(244 348 372 260 364 252 356)(245 349 373 261 365 253 357)(246 350 374 262 366 254 358)(247 351 375 263 367 255 359)(248 352 376 264 368 256 360)(289 393 417 305 409 297 401)(290 394 418 306 410 298 402)(291 395 419 307 411 299 403)(292 396 420 308 412 300 404)(293 397 421 309 413 301 405)(294 398 422 310 414 302 406)(295 399 423 311 415 303 407)(296 400 424 312 416 304 408)
(1 90 13 148)(2 91 14 149)(3 92 15 150)(4 93 16 151)(5 94 9 152)(6 95 10 145)(7 96 11 146)(8 89 12 147)(17 160 183 102)(18 153 184 103)(19 154 177 104)(20 155 178 97)(21 156 179 98)(22 157 180 99)(23 158 181 100)(24 159 182 101)(25 168 191 110)(26 161 192 111)(27 162 185 112)(28 163 186 105)(29 164 187 106)(30 165 188 107)(31 166 189 108)(32 167 190 109)(33 170 199 226)(34 171 200 227)(35 172 193 228)(36 173 194 229)(37 174 195 230)(38 175 196 231)(39 176 197 232)(40 169 198 225)(41 72 207 122)(42 65 208 123)(43 66 201 124)(44 67 202 125)(45 68 203 126)(46 69 204 127)(47 70 205 128)(48 71 206 121)(49 80 215 130)(50 73 216 131)(51 74 209 132)(52 75 210 133)(53 76 211 134)(54 77 212 135)(55 78 213 136)(56 79 214 129)(57 88 223 138)(58 81 224 139)(59 82 217 140)(60 83 218 141)(61 84 219 142)(62 85 220 143)(63 86 221 144)(64 87 222 137)(113 375 239 417)(114 376 240 418)(115 369 233 419)(116 370 234 420)(117 371 235 421)(118 372 236 422)(119 373 237 423)(120 374 238 424)(241 378 291 428)(242 379 292 429)(243 380 293 430)(244 381 294 431)(245 382 295 432)(246 383 296 425)(247 384 289 426)(248 377 290 427)(249 386 299 436)(250 387 300 437)(251 388 301 438)(252 389 302 439)(253 390 303 440)(254 391 304 433)(255 392 297 434)(256 385 298 435)(257 342 307 444)(258 343 308 445)(259 344 309 446)(260 337 310 447)(261 338 311 448)(262 339 312 441)(263 340 305 442)(264 341 306 443)(265 394 315 352)(266 395 316 345)(267 396 317 346)(268 397 318 347)(269 398 319 348)(270 399 320 349)(271 400 313 350)(272 393 314 351)(273 402 323 360)(274 403 324 353)(275 404 325 354)(276 405 326 355)(277 406 327 356)(278 407 328 357)(279 408 321 358)(280 401 322 359)(281 410 331 368)(282 411 332 361)(283 412 333 362)(284 413 334 363)(285 414 335 364)(286 415 336 365)(287 416 329 366)(288 409 330 367)
(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)
(1 352 13 394)(2 397 14 347)(3 350 15 400)(4 395 16 345)(5 348 9 398)(6 393 10 351)(7 346 11 396)(8 399 12 349)(17 406 183 356)(18 359 184 401)(19 404 177 354)(20 357 178 407)(21 402 179 360)(22 355 180 405)(23 408 181 358)(24 353 182 403)(25 414 191 364)(26 367 192 409)(27 412 185 362)(28 365 186 415)(29 410 187 368)(30 363 188 413)(31 416 189 366)(32 361 190 411)(33 422 199 372)(34 375 200 417)(35 420 193 370)(36 373 194 423)(37 418 195 376)(38 371 196 421)(39 424 197 374)(40 369 198 419)(41 430 207 380)(42 383 208 425)(43 428 201 378)(44 381 202 431)(45 426 203 384)(46 379 204 429)(47 432 205 382)(48 377 206 427)(49 438 215 388)(50 391 216 433)(51 436 209 386)(52 389 210 439)(53 434 211 392)(54 387 212 437)(55 440 213 390)(56 385 214 435)(57 446 223 344)(58 339 224 441)(59 444 217 342)(60 337 218 447)(61 442 219 340)(62 343 220 445)(63 448 221 338)(64 341 222 443)(65 246 123 296)(66 291 124 241)(67 244 125 294)(68 289 126 247)(69 242 127 292)(70 295 128 245)(71 248 121 290)(72 293 122 243)(73 254 131 304)(74 299 132 249)(75 252 133 302)(76 297 134 255)(77 250 135 300)(78 303 136 253)(79 256 129 298)(80 301 130 251)(81 262 139 312)(82 307 140 257)(83 260 141 310)(84 305 142 263)(85 258 143 308)(86 311 144 261)(87 264 137 306)(88 309 138 259)(89 270 147 320)(90 315 148 265)(91 268 149 318)(92 313 150 271)(93 266 151 316)(94 319 152 269)(95 272 145 314)(96 317 146 267)(97 278 155 328)(98 323 156 273)(99 276 157 326)(100 321 158 279)(101 274 159 324)(102 327 160 277)(103 280 153 322)(104 325 154 275)(105 286 163 336)(106 331 164 281)(107 284 165 334)(108 329 166 287)(109 282 167 332)(110 335 168 285)(111 288 161 330)(112 333 162 283)(113 227 239 171)(114 174 240 230)(115 225 233 169)(116 172 234 228)(117 231 235 175)(118 170 236 226)(119 229 237 173)(120 176 238 232)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 7A ··· 7F 8A ··· 8H 14A ··· 14R 28A ··· 28AJ 28AK ··· 28BH 56A ··· 56AV order 1 2 2 2 4 ··· 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 ··· 2 8 8 8 8 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C7 C14 C14 C14 D4 SD16 Q16 C4○D4 C7×D4 C7×SD16 C7×Q16 C7×C4○D4 kernel C7×C4.SD16 C4×C56 C7×Q8⋊C4 C7×C4⋊Q8 C4.SD16 C4×C8 Q8⋊C4 C4⋊Q8 C2×C28 C28 C28 C28 C2×C4 C4 C4 C4 # reps 1 1 4 2 6 6 24 12 2 4 4 4 12 24 24 24

Matrix representation of C7×C4.SD16 in GL4(𝔽113) generated by

 49 0 0 0 0 49 0 0 0 0 30 0 0 0 0 30
,
 1 0 0 0 0 1 0 0 0 0 106 83 0 0 77 7
,
 13 13 0 0 100 13 0 0 0 0 8 2 0 0 25 105
,
 82 82 0 0 82 31 0 0 0 0 23 72 0 0 46 90
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,106,77,0,0,83,7],[13,100,0,0,13,13,0,0,0,0,8,25,0,0,2,105],[82,82,0,0,82,31,0,0,0,0,23,46,0,0,72,90] >;

C7×C4.SD16 in GAP, Magma, Sage, TeX

C_7\times C_4.{\rm SD}_{16}
% in TeX

G:=Group("C7xC4.SD16");
// GroupNames label

G:=SmallGroup(448,895);
// by ID

G=gap.SmallGroup(448,895);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,792,2438,310,9804,172,14117,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^4=c^8=1,d^2=b^2,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=b^2*c^3>;
// generators/relations

׿
×
𝔽