direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C4.10D8, C28.61D8, C28.27Q16, C28.52SD16, C4⋊C4.2C28, C4⋊C8.2C14, C4.10(C7×D8), C4⋊Q8.1C14, C4.5(C7×Q16), C4.6(C7×SD16), (C2×C28).504D4, C42.4(C2×C14), (C4×C28).244C22, C14.35(D4⋊C4), C14.16(Q8⋊C4), C14.12(C4.10D4), (C7×C4⋊C4).4C4, (C7×C4⋊C8).8C2, (C7×C4⋊Q8).16C2, (C2×C4).12(C2×C28), (C2×C4).110(C7×D4), C2.5(C7×D4⋊C4), C2.4(C7×Q8⋊C4), (C2×C28).179(C2×C4), C2.4(C7×C4.10D4), C22.40(C7×C22⋊C4), (C2×C14).127(C22⋊C4), SmallGroup(448,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4.10D8
G = < a,b,c,d | a7=b4=c8=1, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=bc-1 >
Subgroups: 114 in 64 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C28, C28, C2×C14, C4⋊C8, C4⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C4.10D8, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C56, Q8×C14, C7×C4⋊C8, C7×C4⋊Q8, C7×C4.10D8
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, D8, SD16, Q16, C28, C2×C14, C4.10D4, D4⋊C4, Q8⋊C4, C2×C28, C7×D4, C4.10D8, C7×C22⋊C4, C7×D8, C7×SD16, C7×Q16, C7×C4.10D4, C7×D4⋊C4, C7×Q8⋊C4, C7×C4.10D8
►Regular action on 448 points
Generators in S448
(1 391 383 375 367 359 351)(2 392 384 376 368 360 352)(3 385 377 369 361 353 345)(4 386 378 370 362 354 346)(5 387 379 371 363 355 347)(6 388 380 372 364 356 348)(7 389 381 373 365 357 349)(8 390 382 374 366 358 350)(9 57 49 41 33 25 17)(10 58 50 42 34 26 18)(11 59 51 43 35 27 19)(12 60 52 44 36 28 20)(13 61 53 45 37 29 21)(14 62 54 46 38 30 22)(15 63 55 47 39 31 23)(16 64 56 48 40 32 24)(65 113 105 97 89 81 73)(66 114 106 98 90 82 74)(67 115 107 99 91 83 75)(68 116 108 100 92 84 76)(69 117 109 101 93 85 77)(70 118 110 102 94 86 78)(71 119 111 103 95 87 79)(72 120 112 104 96 88 80)(121 169 161 153 145 137 129)(122 170 162 154 146 138 130)(123 171 163 155 147 139 131)(124 172 164 156 148 140 132)(125 173 165 157 149 141 133)(126 174 166 158 150 142 134)(127 175 167 159 151 143 135)(128 176 168 160 152 144 136)(177 225 217 209 201 193 185)(178 226 218 210 202 194 186)(179 227 219 211 203 195 187)(180 228 220 212 204 196 188)(181 229 221 213 205 197 189)(182 230 222 214 206 198 190)(183 231 223 215 207 199 191)(184 232 224 216 208 200 192)(233 281 273 265 257 249 241)(234 282 274 266 258 250 242)(235 283 275 267 259 251 243)(236 284 276 268 260 252 244)(237 285 277 269 261 253 245)(238 286 278 270 262 254 246)(239 287 279 271 263 255 247)(240 288 280 272 264 256 248)(289 337 329 321 313 305 297)(290 338 330 322 314 306 298)(291 339 331 323 315 307 299)(292 340 332 324 316 308 300)(293 341 333 325 317 309 301)(294 342 334 326 318 310 302)(295 343 335 327 319 311 303)(296 344 336 328 320 312 304)(393 441 433 425 417 409 401)(394 442 434 426 418 410 402)(395 443 435 427 419 411 403)(396 444 436 428 420 412 404)(397 445 437 429 421 413 405)(398 446 438 430 422 414 406)(399 447 439 431 423 415 407)(400 448 440 432 424 416 408)
(1 121 11 237)(2 238 12 122)(3 123 13 239)(4 240 14 124)(5 125 15 233)(6 234 16 126)(7 127 9 235)(8 236 10 128)(17 243 349 135)(18 136 350 244)(19 245 351 129)(20 130 352 246)(21 247 345 131)(22 132 346 248)(23 241 347 133)(24 134 348 242)(25 251 357 143)(26 144 358 252)(27 253 359 137)(28 138 360 254)(29 255 353 139)(30 140 354 256)(31 249 355 141)(32 142 356 250)(33 259 365 151)(34 152 366 260)(35 261 367 145)(36 146 368 262)(37 263 361 147)(38 148 362 264)(39 257 363 149)(40 150 364 258)(41 267 373 159)(42 160 374 268)(43 269 375 153)(44 154 376 270)(45 271 369 155)(46 156 370 272)(47 265 371 157)(48 158 372 266)(49 275 381 167)(50 168 382 276)(51 277 383 161)(52 162 384 278)(53 279 377 163)(54 164 378 280)(55 273 379 165)(56 166 380 274)(57 283 389 175)(58 176 390 284)(59 285 391 169)(60 170 392 286)(61 287 385 171)(62 172 386 288)(63 281 387 173)(64 174 388 282)(65 179 397 295)(66 296 398 180)(67 181 399 289)(68 290 400 182)(69 183 393 291)(70 292 394 184)(71 177 395 293)(72 294 396 178)(73 187 405 303)(74 304 406 188)(75 189 407 297)(76 298 408 190)(77 191 401 299)(78 300 402 192)(79 185 403 301)(80 302 404 186)(81 195 413 311)(82 312 414 196)(83 197 415 305)(84 306 416 198)(85 199 409 307)(86 308 410 200)(87 193 411 309)(88 310 412 194)(89 203 421 319)(90 320 422 204)(91 205 423 313)(92 314 424 206)(93 207 417 315)(94 316 418 208)(95 201 419 317)(96 318 420 202)(97 211 429 327)(98 328 430 212)(99 213 431 321)(100 322 432 214)(101 215 425 323)(102 324 426 216)(103 209 427 325)(104 326 428 210)(105 219 437 335)(106 336 438 220)(107 221 439 329)(108 330 440 222)(109 223 433 331)(110 332 434 224)(111 217 435 333)(112 334 436 218)(113 227 445 343)(114 344 446 228)(115 229 447 337)(116 338 448 230)(117 231 441 339)(118 340 442 232)(119 225 443 341)(120 342 444 226)
(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)
(1 292 237 70 11 184 121 394)(2 69 122 291 12 393 238 183)(3 290 239 68 13 182 123 400)(4 67 124 289 14 399 240 181)(5 296 233 66 15 180 125 398)(6 65 126 295 16 397 234 179)(7 294 235 72 9 178 127 396)(8 71 128 293 10 395 236 177)(17 186 135 404 349 302 243 80)(18 403 244 185 350 79 136 301)(19 192 129 402 351 300 245 78)(20 401 246 191 352 77 130 299)(21 190 131 408 345 298 247 76)(22 407 248 189 346 75 132 297)(23 188 133 406 347 304 241 74)(24 405 242 187 348 73 134 303)(25 194 143 412 357 310 251 88)(26 411 252 193 358 87 144 309)(27 200 137 410 359 308 253 86)(28 409 254 199 360 85 138 307)(29 198 139 416 353 306 255 84)(30 415 256 197 354 83 140 305)(31 196 141 414 355 312 249 82)(32 413 250 195 356 81 142 311)(33 202 151 420 365 318 259 96)(34 419 260 201 366 95 152 317)(35 208 145 418 367 316 261 94)(36 417 262 207 368 93 146 315)(37 206 147 424 361 314 263 92)(38 423 264 205 362 91 148 313)(39 204 149 422 363 320 257 90)(40 421 258 203 364 89 150 319)(41 210 159 428 373 326 267 104)(42 427 268 209 374 103 160 325)(43 216 153 426 375 324 269 102)(44 425 270 215 376 101 154 323)(45 214 155 432 369 322 271 100)(46 431 272 213 370 99 156 321)(47 212 157 430 371 328 265 98)(48 429 266 211 372 97 158 327)(49 218 167 436 381 334 275 112)(50 435 276 217 382 111 168 333)(51 224 161 434 383 332 277 110)(52 433 278 223 384 109 162 331)(53 222 163 440 377 330 279 108)(54 439 280 221 378 107 164 329)(55 220 165 438 379 336 273 106)(56 437 274 219 380 105 166 335)(57 226 175 444 389 342 283 120)(58 443 284 225 390 119 176 341)(59 232 169 442 391 340 285 118)(60 441 286 231 392 117 170 339)(61 230 171 448 385 338 287 116)(62 447 288 229 386 115 172 337)(63 228 173 446 387 344 281 114)(64 445 282 227 388 113 174 343)
G:=sub<Sym(448)| (1,391,383,375,367,359,351)(2,392,384,376,368,360,352)(3,385,377,369,361,353,345)(4,386,378,370,362,354,346)(5,387,379,371,363,355,347)(6,388,380,372,364,356,348)(7,389,381,373,365,357,349)(8,390,382,374,366,358,350)(9,57,49,41,33,25,17)(10,58,50,42,34,26,18)(11,59,51,43,35,27,19)(12,60,52,44,36,28,20)(13,61,53,45,37,29,21)(14,62,54,46,38,30,22)(15,63,55,47,39,31,23)(16,64,56,48,40,32,24)(65,113,105,97,89,81,73)(66,114,106,98,90,82,74)(67,115,107,99,91,83,75)(68,116,108,100,92,84,76)(69,117,109,101,93,85,77)(70,118,110,102,94,86,78)(71,119,111,103,95,87,79)(72,120,112,104,96,88,80)(121,169,161,153,145,137,129)(122,170,162,154,146,138,130)(123,171,163,155,147,139,131)(124,172,164,156,148,140,132)(125,173,165,157,149,141,133)(126,174,166,158,150,142,134)(127,175,167,159,151,143,135)(128,176,168,160,152,144,136)(177,225,217,209,201,193,185)(178,226,218,210,202,194,186)(179,227,219,211,203,195,187)(180,228,220,212,204,196,188)(181,229,221,213,205,197,189)(182,230,222,214,206,198,190)(183,231,223,215,207,199,191)(184,232,224,216,208,200,192)(233,281,273,265,257,249,241)(234,282,274,266,258,250,242)(235,283,275,267,259,251,243)(236,284,276,268,260,252,244)(237,285,277,269,261,253,245)(238,286,278,270,262,254,246)(239,287,279,271,263,255,247)(240,288,280,272,264,256,248)(289,337,329,321,313,305,297)(290,338,330,322,314,306,298)(291,339,331,323,315,307,299)(292,340,332,324,316,308,300)(293,341,333,325,317,309,301)(294,342,334,326,318,310,302)(295,343,335,327,319,311,303)(296,344,336,328,320,312,304)(393,441,433,425,417,409,401)(394,442,434,426,418,410,402)(395,443,435,427,419,411,403)(396,444,436,428,420,412,404)(397,445,437,429,421,413,405)(398,446,438,430,422,414,406)(399,447,439,431,423,415,407)(400,448,440,432,424,416,408), (1,121,11,237)(2,238,12,122)(3,123,13,239)(4,240,14,124)(5,125,15,233)(6,234,16,126)(7,127,9,235)(8,236,10,128)(17,243,349,135)(18,136,350,244)(19,245,351,129)(20,130,352,246)(21,247,345,131)(22,132,346,248)(23,241,347,133)(24,134,348,242)(25,251,357,143)(26,144,358,252)(27,253,359,137)(28,138,360,254)(29,255,353,139)(30,140,354,256)(31,249,355,141)(32,142,356,250)(33,259,365,151)(34,152,366,260)(35,261,367,145)(36,146,368,262)(37,263,361,147)(38,148,362,264)(39,257,363,149)(40,150,364,258)(41,267,373,159)(42,160,374,268)(43,269,375,153)(44,154,376,270)(45,271,369,155)(46,156,370,272)(47,265,371,157)(48,158,372,266)(49,275,381,167)(50,168,382,276)(51,277,383,161)(52,162,384,278)(53,279,377,163)(54,164,378,280)(55,273,379,165)(56,166,380,274)(57,283,389,175)(58,176,390,284)(59,285,391,169)(60,170,392,286)(61,287,385,171)(62,172,386,288)(63,281,387,173)(64,174,388,282)(65,179,397,295)(66,296,398,180)(67,181,399,289)(68,290,400,182)(69,183,393,291)(70,292,394,184)(71,177,395,293)(72,294,396,178)(73,187,405,303)(74,304,406,188)(75,189,407,297)(76,298,408,190)(77,191,401,299)(78,300,402,192)(79,185,403,301)(80,302,404,186)(81,195,413,311)(82,312,414,196)(83,197,415,305)(84,306,416,198)(85,199,409,307)(86,308,410,200)(87,193,411,309)(88,310,412,194)(89,203,421,319)(90,320,422,204)(91,205,423,313)(92,314,424,206)(93,207,417,315)(94,316,418,208)(95,201,419,317)(96,318,420,202)(97,211,429,327)(98,328,430,212)(99,213,431,321)(100,322,432,214)(101,215,425,323)(102,324,426,216)(103,209,427,325)(104,326,428,210)(105,219,437,335)(106,336,438,220)(107,221,439,329)(108,330,440,222)(109,223,433,331)(110,332,434,224)(111,217,435,333)(112,334,436,218)(113,227,445,343)(114,344,446,228)(115,229,447,337)(116,338,448,230)(117,231,441,339)(118,340,442,232)(119,225,443,341)(120,342,444,226), (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), (1,292,237,70,11,184,121,394)(2,69,122,291,12,393,238,183)(3,290,239,68,13,182,123,400)(4,67,124,289,14,399,240,181)(5,296,233,66,15,180,125,398)(6,65,126,295,16,397,234,179)(7,294,235,72,9,178,127,396)(8,71,128,293,10,395,236,177)(17,186,135,404,349,302,243,80)(18,403,244,185,350,79,136,301)(19,192,129,402,351,300,245,78)(20,401,246,191,352,77,130,299)(21,190,131,408,345,298,247,76)(22,407,248,189,346,75,132,297)(23,188,133,406,347,304,241,74)(24,405,242,187,348,73,134,303)(25,194,143,412,357,310,251,88)(26,411,252,193,358,87,144,309)(27,200,137,410,359,308,253,86)(28,409,254,199,360,85,138,307)(29,198,139,416,353,306,255,84)(30,415,256,197,354,83,140,305)(31,196,141,414,355,312,249,82)(32,413,250,195,356,81,142,311)(33,202,151,420,365,318,259,96)(34,419,260,201,366,95,152,317)(35,208,145,418,367,316,261,94)(36,417,262,207,368,93,146,315)(37,206,147,424,361,314,263,92)(38,423,264,205,362,91,148,313)(39,204,149,422,363,320,257,90)(40,421,258,203,364,89,150,319)(41,210,159,428,373,326,267,104)(42,427,268,209,374,103,160,325)(43,216,153,426,375,324,269,102)(44,425,270,215,376,101,154,323)(45,214,155,432,369,322,271,100)(46,431,272,213,370,99,156,321)(47,212,157,430,371,328,265,98)(48,429,266,211,372,97,158,327)(49,218,167,436,381,334,275,112)(50,435,276,217,382,111,168,333)(51,224,161,434,383,332,277,110)(52,433,278,223,384,109,162,331)(53,222,163,440,377,330,279,108)(54,439,280,221,378,107,164,329)(55,220,165,438,379,336,273,106)(56,437,274,219,380,105,166,335)(57,226,175,444,389,342,283,120)(58,443,284,225,390,119,176,341)(59,232,169,442,391,340,285,118)(60,441,286,231,392,117,170,339)(61,230,171,448,385,338,287,116)(62,447,288,229,386,115,172,337)(63,228,173,446,387,344,281,114)(64,445,282,227,388,113,174,343)>;
G:=Group( (1,391,383,375,367,359,351)(2,392,384,376,368,360,352)(3,385,377,369,361,353,345)(4,386,378,370,362,354,346)(5,387,379,371,363,355,347)(6,388,380,372,364,356,348)(7,389,381,373,365,357,349)(8,390,382,374,366,358,350)(9,57,49,41,33,25,17)(10,58,50,42,34,26,18)(11,59,51,43,35,27,19)(12,60,52,44,36,28,20)(13,61,53,45,37,29,21)(14,62,54,46,38,30,22)(15,63,55,47,39,31,23)(16,64,56,48,40,32,24)(65,113,105,97,89,81,73)(66,114,106,98,90,82,74)(67,115,107,99,91,83,75)(68,116,108,100,92,84,76)(69,117,109,101,93,85,77)(70,118,110,102,94,86,78)(71,119,111,103,95,87,79)(72,120,112,104,96,88,80)(121,169,161,153,145,137,129)(122,170,162,154,146,138,130)(123,171,163,155,147,139,131)(124,172,164,156,148,140,132)(125,173,165,157,149,141,133)(126,174,166,158,150,142,134)(127,175,167,159,151,143,135)(128,176,168,160,152,144,136)(177,225,217,209,201,193,185)(178,226,218,210,202,194,186)(179,227,219,211,203,195,187)(180,228,220,212,204,196,188)(181,229,221,213,205,197,189)(182,230,222,214,206,198,190)(183,231,223,215,207,199,191)(184,232,224,216,208,200,192)(233,281,273,265,257,249,241)(234,282,274,266,258,250,242)(235,283,275,267,259,251,243)(236,284,276,268,260,252,244)(237,285,277,269,261,253,245)(238,286,278,270,262,254,246)(239,287,279,271,263,255,247)(240,288,280,272,264,256,248)(289,337,329,321,313,305,297)(290,338,330,322,314,306,298)(291,339,331,323,315,307,299)(292,340,332,324,316,308,300)(293,341,333,325,317,309,301)(294,342,334,326,318,310,302)(295,343,335,327,319,311,303)(296,344,336,328,320,312,304)(393,441,433,425,417,409,401)(394,442,434,426,418,410,402)(395,443,435,427,419,411,403)(396,444,436,428,420,412,404)(397,445,437,429,421,413,405)(398,446,438,430,422,414,406)(399,447,439,431,423,415,407)(400,448,440,432,424,416,408), (1,121,11,237)(2,238,12,122)(3,123,13,239)(4,240,14,124)(5,125,15,233)(6,234,16,126)(7,127,9,235)(8,236,10,128)(17,243,349,135)(18,136,350,244)(19,245,351,129)(20,130,352,246)(21,247,345,131)(22,132,346,248)(23,241,347,133)(24,134,348,242)(25,251,357,143)(26,144,358,252)(27,253,359,137)(28,138,360,254)(29,255,353,139)(30,140,354,256)(31,249,355,141)(32,142,356,250)(33,259,365,151)(34,152,366,260)(35,261,367,145)(36,146,368,262)(37,263,361,147)(38,148,362,264)(39,257,363,149)(40,150,364,258)(41,267,373,159)(42,160,374,268)(43,269,375,153)(44,154,376,270)(45,271,369,155)(46,156,370,272)(47,265,371,157)(48,158,372,266)(49,275,381,167)(50,168,382,276)(51,277,383,161)(52,162,384,278)(53,279,377,163)(54,164,378,280)(55,273,379,165)(56,166,380,274)(57,283,389,175)(58,176,390,284)(59,285,391,169)(60,170,392,286)(61,287,385,171)(62,172,386,288)(63,281,387,173)(64,174,388,282)(65,179,397,295)(66,296,398,180)(67,181,399,289)(68,290,400,182)(69,183,393,291)(70,292,394,184)(71,177,395,293)(72,294,396,178)(73,187,405,303)(74,304,406,188)(75,189,407,297)(76,298,408,190)(77,191,401,299)(78,300,402,192)(79,185,403,301)(80,302,404,186)(81,195,413,311)(82,312,414,196)(83,197,415,305)(84,306,416,198)(85,199,409,307)(86,308,410,200)(87,193,411,309)(88,310,412,194)(89,203,421,319)(90,320,422,204)(91,205,423,313)(92,314,424,206)(93,207,417,315)(94,316,418,208)(95,201,419,317)(96,318,420,202)(97,211,429,327)(98,328,430,212)(99,213,431,321)(100,322,432,214)(101,215,425,323)(102,324,426,216)(103,209,427,325)(104,326,428,210)(105,219,437,335)(106,336,438,220)(107,221,439,329)(108,330,440,222)(109,223,433,331)(110,332,434,224)(111,217,435,333)(112,334,436,218)(113,227,445,343)(114,344,446,228)(115,229,447,337)(116,338,448,230)(117,231,441,339)(118,340,442,232)(119,225,443,341)(120,342,444,226), (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), (1,292,237,70,11,184,121,394)(2,69,122,291,12,393,238,183)(3,290,239,68,13,182,123,400)(4,67,124,289,14,399,240,181)(5,296,233,66,15,180,125,398)(6,65,126,295,16,397,234,179)(7,294,235,72,9,178,127,396)(8,71,128,293,10,395,236,177)(17,186,135,404,349,302,243,80)(18,403,244,185,350,79,136,301)(19,192,129,402,351,300,245,78)(20,401,246,191,352,77,130,299)(21,190,131,408,345,298,247,76)(22,407,248,189,346,75,132,297)(23,188,133,406,347,304,241,74)(24,405,242,187,348,73,134,303)(25,194,143,412,357,310,251,88)(26,411,252,193,358,87,144,309)(27,200,137,410,359,308,253,86)(28,409,254,199,360,85,138,307)(29,198,139,416,353,306,255,84)(30,415,256,197,354,83,140,305)(31,196,141,414,355,312,249,82)(32,413,250,195,356,81,142,311)(33,202,151,420,365,318,259,96)(34,419,260,201,366,95,152,317)(35,208,145,418,367,316,261,94)(36,417,262,207,368,93,146,315)(37,206,147,424,361,314,263,92)(38,423,264,205,362,91,148,313)(39,204,149,422,363,320,257,90)(40,421,258,203,364,89,150,319)(41,210,159,428,373,326,267,104)(42,427,268,209,374,103,160,325)(43,216,153,426,375,324,269,102)(44,425,270,215,376,101,154,323)(45,214,155,432,369,322,271,100)(46,431,272,213,370,99,156,321)(47,212,157,430,371,328,265,98)(48,429,266,211,372,97,158,327)(49,218,167,436,381,334,275,112)(50,435,276,217,382,111,168,333)(51,224,161,434,383,332,277,110)(52,433,278,223,384,109,162,331)(53,222,163,440,377,330,279,108)(54,439,280,221,378,107,164,329)(55,220,165,438,379,336,273,106)(56,437,274,219,380,105,166,335)(57,226,175,444,389,342,283,120)(58,443,284,225,390,119,176,341)(59,232,169,442,391,340,285,118)(60,441,286,231,392,117,170,339)(61,230,171,448,385,338,287,116)(62,447,288,229,386,115,172,337)(63,228,173,446,387,344,281,114)(64,445,282,227,388,113,174,343) );
G=PermutationGroup([[(1,391,383,375,367,359,351),(2,392,384,376,368,360,352),(3,385,377,369,361,353,345),(4,386,378,370,362,354,346),(5,387,379,371,363,355,347),(6,388,380,372,364,356,348),(7,389,381,373,365,357,349),(8,390,382,374,366,358,350),(9,57,49,41,33,25,17),(10,58,50,42,34,26,18),(11,59,51,43,35,27,19),(12,60,52,44,36,28,20),(13,61,53,45,37,29,21),(14,62,54,46,38,30,22),(15,63,55,47,39,31,23),(16,64,56,48,40,32,24),(65,113,105,97,89,81,73),(66,114,106,98,90,82,74),(67,115,107,99,91,83,75),(68,116,108,100,92,84,76),(69,117,109,101,93,85,77),(70,118,110,102,94,86,78),(71,119,111,103,95,87,79),(72,120,112,104,96,88,80),(121,169,161,153,145,137,129),(122,170,162,154,146,138,130),(123,171,163,155,147,139,131),(124,172,164,156,148,140,132),(125,173,165,157,149,141,133),(126,174,166,158,150,142,134),(127,175,167,159,151,143,135),(128,176,168,160,152,144,136),(177,225,217,209,201,193,185),(178,226,218,210,202,194,186),(179,227,219,211,203,195,187),(180,228,220,212,204,196,188),(181,229,221,213,205,197,189),(182,230,222,214,206,198,190),(183,231,223,215,207,199,191),(184,232,224,216,208,200,192),(233,281,273,265,257,249,241),(234,282,274,266,258,250,242),(235,283,275,267,259,251,243),(236,284,276,268,260,252,244),(237,285,277,269,261,253,245),(238,286,278,270,262,254,246),(239,287,279,271,263,255,247),(240,288,280,272,264,256,248),(289,337,329,321,313,305,297),(290,338,330,322,314,306,298),(291,339,331,323,315,307,299),(292,340,332,324,316,308,300),(293,341,333,325,317,309,301),(294,342,334,326,318,310,302),(295,343,335,327,319,311,303),(296,344,336,328,320,312,304),(393,441,433,425,417,409,401),(394,442,434,426,418,410,402),(395,443,435,427,419,411,403),(396,444,436,428,420,412,404),(397,445,437,429,421,413,405),(398,446,438,430,422,414,406),(399,447,439,431,423,415,407),(400,448,440,432,424,416,408)], [(1,121,11,237),(2,238,12,122),(3,123,13,239),(4,240,14,124),(5,125,15,233),(6,234,16,126),(7,127,9,235),(8,236,10,128),(17,243,349,135),(18,136,350,244),(19,245,351,129),(20,130,352,246),(21,247,345,131),(22,132,346,248),(23,241,347,133),(24,134,348,242),(25,251,357,143),(26,144,358,252),(27,253,359,137),(28,138,360,254),(29,255,353,139),(30,140,354,256),(31,249,355,141),(32,142,356,250),(33,259,365,151),(34,152,366,260),(35,261,367,145),(36,146,368,262),(37,263,361,147),(38,148,362,264),(39,257,363,149),(40,150,364,258),(41,267,373,159),(42,160,374,268),(43,269,375,153),(44,154,376,270),(45,271,369,155),(46,156,370,272),(47,265,371,157),(48,158,372,266),(49,275,381,167),(50,168,382,276),(51,277,383,161),(52,162,384,278),(53,279,377,163),(54,164,378,280),(55,273,379,165),(56,166,380,274),(57,283,389,175),(58,176,390,284),(59,285,391,169),(60,170,392,286),(61,287,385,171),(62,172,386,288),(63,281,387,173),(64,174,388,282),(65,179,397,295),(66,296,398,180),(67,181,399,289),(68,290,400,182),(69,183,393,291),(70,292,394,184),(71,177,395,293),(72,294,396,178),(73,187,405,303),(74,304,406,188),(75,189,407,297),(76,298,408,190),(77,191,401,299),(78,300,402,192),(79,185,403,301),(80,302,404,186),(81,195,413,311),(82,312,414,196),(83,197,415,305),(84,306,416,198),(85,199,409,307),(86,308,410,200),(87,193,411,309),(88,310,412,194),(89,203,421,319),(90,320,422,204),(91,205,423,313),(92,314,424,206),(93,207,417,315),(94,316,418,208),(95,201,419,317),(96,318,420,202),(97,211,429,327),(98,328,430,212),(99,213,431,321),(100,322,432,214),(101,215,425,323),(102,324,426,216),(103,209,427,325),(104,326,428,210),(105,219,437,335),(106,336,438,220),(107,221,439,329),(108,330,440,222),(109,223,433,331),(110,332,434,224),(111,217,435,333),(112,334,436,218),(113,227,445,343),(114,344,446,228),(115,229,447,337),(116,338,448,230),(117,231,441,339),(118,340,442,232),(119,225,443,341),(120,342,444,226)], [(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)], [(1,292,237,70,11,184,121,394),(2,69,122,291,12,393,238,183),(3,290,239,68,13,182,123,400),(4,67,124,289,14,399,240,181),(5,296,233,66,15,180,125,398),(6,65,126,295,16,397,234,179),(7,294,235,72,9,178,127,396),(8,71,128,293,10,395,236,177),(17,186,135,404,349,302,243,80),(18,403,244,185,350,79,136,301),(19,192,129,402,351,300,245,78),(20,401,246,191,352,77,130,299),(21,190,131,408,345,298,247,76),(22,407,248,189,346,75,132,297),(23,188,133,406,347,304,241,74),(24,405,242,187,348,73,134,303),(25,194,143,412,357,310,251,88),(26,411,252,193,358,87,144,309),(27,200,137,410,359,308,253,86),(28,409,254,199,360,85,138,307),(29,198,139,416,353,306,255,84),(30,415,256,197,354,83,140,305),(31,196,141,414,355,312,249,82),(32,413,250,195,356,81,142,311),(33,202,151,420,365,318,259,96),(34,419,260,201,366,95,152,317),(35,208,145,418,367,316,261,94),(36,417,262,207,368,93,146,315),(37,206,147,424,361,314,263,92),(38,423,264,205,362,91,148,313),(39,204,149,422,363,320,257,90),(40,421,258,203,364,89,150,319),(41,210,159,428,373,326,267,104),(42,427,268,209,374,103,160,325),(43,216,153,426,375,324,269,102),(44,425,270,215,376,101,154,323),(45,214,155,432,369,322,271,100),(46,431,272,213,370,99,156,321),(47,212,157,430,371,328,265,98),(48,429,266,211,372,97,158,327),(49,218,167,436,381,334,275,112),(50,435,276,217,382,111,168,333),(51,224,161,434,383,332,277,110),(52,433,278,223,384,109,162,331),(53,222,163,440,377,330,279,108),(54,439,280,221,378,107,164,329),(55,220,165,438,379,336,273,106),(56,437,274,219,380,105,166,335),(57,226,175,444,389,342,283,120),(58,443,284,225,390,119,176,341),(59,232,169,442,391,340,285,118),(60,441,286,231,392,117,170,339),(61,230,171,448,385,338,287,116),(62,447,288,229,386,115,172,337),(63,228,173,446,387,344,281,114),(64,445,282,227,388,113,174,343)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 28A | ··· | 28X | 28Y | ··· | 28AD | 28AE | ··· | 28AP | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | D8 | SD16 | Q16 | C7×D4 | C7×D8 | C7×SD16 | C7×Q16 | C4.10D4 | C7×C4.10D4 |
| kernel | C7×C4.10D8 | C7×C4⋊C8 | C7×C4⋊Q8 | C7×C4⋊C4 | C4.10D8 | C4⋊C8 | C4⋊Q8 | C4⋊C4 | C2×C28 | C28 | C28 | C28 | C2×C4 | C4 | C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 2 | 2 | 4 | 2 | 12 | 12 | 24 | 12 | 1 | 6 |
Matrix representation of C7×C4.10D8 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 53 | 93 | 0 | 0 |
| 50 | 60 | 0 | 0 |
| 0 | 0 | 31 | 31 |
| 0 | 0 | 82 | 31 |
| 87 | 87 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 102 | 85 |
| 0 | 0 | 85 | 11 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,1,0,0,0,0,1],[1,112,0,0,2,112,0,0,0,0,1,0,0,0,0,1],[53,50,0,0,93,60,0,0,0,0,31,82,0,0,31,31],[87,13,0,0,87,0,0,0,0,0,102,85,0,0,85,11] >;
C7×C4.10D8 in GAP, Magma, Sage, TeX
C_7\times C_4._{10}D_8 % in TeX
G:=Group("C7xC4.10D8"); // GroupNames label
G:=SmallGroup(448,136);
// by ID
G=gap.SmallGroup(448,136);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,3923,3538,248,6871,242]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^8=1,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations