direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C31, C4⋊C124, C124⋊3C4, C62.3Q8, C62.13D4, C2.(Q8×C31), (C2×C4).1C62, C2.2(D4×C31), C62.11(C2×C4), C2.2(C2×C124), (C2×C124).2C2, C22.3(C2×C62), (C2×C62).14C22, SmallGroup(496,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C31
G = < a,b,c | a31=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4⋊C4×C31
►Regular action on 496 points
Generators in S496
(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)(435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465)(466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
(1 182 360 87)(2 183 361 88)(3 184 362 89)(4 185 363 90)(5 186 364 91)(6 156 365 92)(7 157 366 93)(8 158 367 63)(9 159 368 64)(10 160 369 65)(11 161 370 66)(12 162 371 67)(13 163 372 68)(14 164 342 69)(15 165 343 70)(16 166 344 71)(17 167 345 72)(18 168 346 73)(19 169 347 74)(20 170 348 75)(21 171 349 76)(22 172 350 77)(23 173 351 78)(24 174 352 79)(25 175 353 80)(26 176 354 81)(27 177 355 82)(28 178 356 83)(29 179 357 84)(30 180 358 85)(31 181 359 86)(32 300 148 229)(33 301 149 230)(34 302 150 231)(35 303 151 232)(36 304 152 233)(37 305 153 234)(38 306 154 235)(39 307 155 236)(40 308 125 237)(41 309 126 238)(42 310 127 239)(43 280 128 240)(44 281 129 241)(45 282 130 242)(46 283 131 243)(47 284 132 244)(48 285 133 245)(49 286 134 246)(50 287 135 247)(51 288 136 248)(52 289 137 218)(53 290 138 219)(54 291 139 220)(55 292 140 221)(56 293 141 222)(57 294 142 223)(58 295 143 224)(59 296 144 225)(60 297 145 226)(61 298 146 227)(62 299 147 228)(94 396 465 266)(95 397 435 267)(96 398 436 268)(97 399 437 269)(98 400 438 270)(99 401 439 271)(100 402 440 272)(101 403 441 273)(102 373 442 274)(103 374 443 275)(104 375 444 276)(105 376 445 277)(106 377 446 278)(107 378 447 279)(108 379 448 249)(109 380 449 250)(110 381 450 251)(111 382 451 252)(112 383 452 253)(113 384 453 254)(114 385 454 255)(115 386 455 256)(116 387 456 257)(117 388 457 258)(118 389 458 259)(119 390 459 260)(120 391 460 261)(121 392 461 262)(122 393 462 263)(123 394 463 264)(124 395 464 265)(187 410 495 341)(188 411 496 311)(189 412 466 312)(190 413 467 313)(191 414 468 314)(192 415 469 315)(193 416 470 316)(194 417 471 317)(195 418 472 318)(196 419 473 319)(197 420 474 320)(198 421 475 321)(199 422 476 322)(200 423 477 323)(201 424 478 324)(202 425 479 325)(203 426 480 326)(204 427 481 327)(205 428 482 328)(206 429 483 329)(207 430 484 330)(208 431 485 331)(209 432 486 332)(210 433 487 333)(211 434 488 334)(212 404 489 335)(213 405 490 336)(214 406 491 337)(215 407 492 338)(216 408 493 339)(217 409 494 340)
(1 197 227 382)(2 198 228 383)(3 199 229 384)(4 200 230 385)(5 201 231 386)(6 202 232 387)(7 203 233 388)(8 204 234 389)(9 205 235 390)(10 206 236 391)(11 207 237 392)(12 208 238 393)(13 209 239 394)(14 210 240 395)(15 211 241 396)(16 212 242 397)(17 213 243 398)(18 214 244 399)(19 215 245 400)(20 216 246 401)(21 217 247 402)(22 187 248 403)(23 188 218 373)(24 189 219 374)(25 190 220 375)(26 191 221 376)(27 192 222 377)(28 193 223 378)(29 194 224 379)(30 195 225 380)(31 196 226 381)(32 113 184 322)(33 114 185 323)(34 115 186 324)(35 116 156 325)(36 117 157 326)(37 118 158 327)(38 119 159 328)(39 120 160 329)(40 121 161 330)(41 122 162 331)(42 123 163 332)(43 124 164 333)(44 94 165 334)(45 95 166 335)(46 96 167 336)(47 97 168 337)(48 98 169 338)(49 99 170 339)(50 100 171 340)(51 101 172 341)(52 102 173 311)(53 103 174 312)(54 104 175 313)(55 105 176 314)(56 106 177 315)(57 107 178 316)(58 108 179 317)(59 109 180 318)(60 110 181 319)(61 111 182 320)(62 112 183 321)(63 427 153 458)(64 428 154 459)(65 429 155 460)(66 430 125 461)(67 431 126 462)(68 432 127 463)(69 433 128 464)(70 434 129 465)(71 404 130 435)(72 405 131 436)(73 406 132 437)(74 407 133 438)(75 408 134 439)(76 409 135 440)(77 410 136 441)(78 411 137 442)(79 412 138 443)(80 413 139 444)(81 414 140 445)(82 415 141 446)(83 416 142 447)(84 417 143 448)(85 418 144 449)(86 419 145 450)(87 420 146 451)(88 421 147 452)(89 422 148 453)(90 423 149 454)(91 424 150 455)(92 425 151 456)(93 426 152 457)(249 357 471 295)(250 358 472 296)(251 359 473 297)(252 360 474 298)(253 361 475 299)(254 362 476 300)(255 363 477 301)(256 364 478 302)(257 365 479 303)(258 366 480 304)(259 367 481 305)(260 368 482 306)(261 369 483 307)(262 370 484 308)(263 371 485 309)(264 372 486 310)(265 342 487 280)(266 343 488 281)(267 344 489 282)(268 345 490 283)(269 346 491 284)(270 347 492 285)(271 348 493 286)(272 349 494 287)(273 350 495 288)(274 351 496 289)(275 352 466 290)(276 353 467 291)(277 354 468 292)(278 355 469 293)(279 356 470 294)
G:=sub<Sym(496)| (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)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465)(466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,182,360,87)(2,183,361,88)(3,184,362,89)(4,185,363,90)(5,186,364,91)(6,156,365,92)(7,157,366,93)(8,158,367,63)(9,159,368,64)(10,160,369,65)(11,161,370,66)(12,162,371,67)(13,163,372,68)(14,164,342,69)(15,165,343,70)(16,166,344,71)(17,167,345,72)(18,168,346,73)(19,169,347,74)(20,170,348,75)(21,171,349,76)(22,172,350,77)(23,173,351,78)(24,174,352,79)(25,175,353,80)(26,176,354,81)(27,177,355,82)(28,178,356,83)(29,179,357,84)(30,180,358,85)(31,181,359,86)(32,300,148,229)(33,301,149,230)(34,302,150,231)(35,303,151,232)(36,304,152,233)(37,305,153,234)(38,306,154,235)(39,307,155,236)(40,308,125,237)(41,309,126,238)(42,310,127,239)(43,280,128,240)(44,281,129,241)(45,282,130,242)(46,283,131,243)(47,284,132,244)(48,285,133,245)(49,286,134,246)(50,287,135,247)(51,288,136,248)(52,289,137,218)(53,290,138,219)(54,291,139,220)(55,292,140,221)(56,293,141,222)(57,294,142,223)(58,295,143,224)(59,296,144,225)(60,297,145,226)(61,298,146,227)(62,299,147,228)(94,396,465,266)(95,397,435,267)(96,398,436,268)(97,399,437,269)(98,400,438,270)(99,401,439,271)(100,402,440,272)(101,403,441,273)(102,373,442,274)(103,374,443,275)(104,375,444,276)(105,376,445,277)(106,377,446,278)(107,378,447,279)(108,379,448,249)(109,380,449,250)(110,381,450,251)(111,382,451,252)(112,383,452,253)(113,384,453,254)(114,385,454,255)(115,386,455,256)(116,387,456,257)(117,388,457,258)(118,389,458,259)(119,390,459,260)(120,391,460,261)(121,392,461,262)(122,393,462,263)(123,394,463,264)(124,395,464,265)(187,410,495,341)(188,411,496,311)(189,412,466,312)(190,413,467,313)(191,414,468,314)(192,415,469,315)(193,416,470,316)(194,417,471,317)(195,418,472,318)(196,419,473,319)(197,420,474,320)(198,421,475,321)(199,422,476,322)(200,423,477,323)(201,424,478,324)(202,425,479,325)(203,426,480,326)(204,427,481,327)(205,428,482,328)(206,429,483,329)(207,430,484,330)(208,431,485,331)(209,432,486,332)(210,433,487,333)(211,434,488,334)(212,404,489,335)(213,405,490,336)(214,406,491,337)(215,407,492,338)(216,408,493,339)(217,409,494,340), (1,197,227,382)(2,198,228,383)(3,199,229,384)(4,200,230,385)(5,201,231,386)(6,202,232,387)(7,203,233,388)(8,204,234,389)(9,205,235,390)(10,206,236,391)(11,207,237,392)(12,208,238,393)(13,209,239,394)(14,210,240,395)(15,211,241,396)(16,212,242,397)(17,213,243,398)(18,214,244,399)(19,215,245,400)(20,216,246,401)(21,217,247,402)(22,187,248,403)(23,188,218,373)(24,189,219,374)(25,190,220,375)(26,191,221,376)(27,192,222,377)(28,193,223,378)(29,194,224,379)(30,195,225,380)(31,196,226,381)(32,113,184,322)(33,114,185,323)(34,115,186,324)(35,116,156,325)(36,117,157,326)(37,118,158,327)(38,119,159,328)(39,120,160,329)(40,121,161,330)(41,122,162,331)(42,123,163,332)(43,124,164,333)(44,94,165,334)(45,95,166,335)(46,96,167,336)(47,97,168,337)(48,98,169,338)(49,99,170,339)(50,100,171,340)(51,101,172,341)(52,102,173,311)(53,103,174,312)(54,104,175,313)(55,105,176,314)(56,106,177,315)(57,107,178,316)(58,108,179,317)(59,109,180,318)(60,110,181,319)(61,111,182,320)(62,112,183,321)(63,427,153,458)(64,428,154,459)(65,429,155,460)(66,430,125,461)(67,431,126,462)(68,432,127,463)(69,433,128,464)(70,434,129,465)(71,404,130,435)(72,405,131,436)(73,406,132,437)(74,407,133,438)(75,408,134,439)(76,409,135,440)(77,410,136,441)(78,411,137,442)(79,412,138,443)(80,413,139,444)(81,414,140,445)(82,415,141,446)(83,416,142,447)(84,417,143,448)(85,418,144,449)(86,419,145,450)(87,420,146,451)(88,421,147,452)(89,422,148,453)(90,423,149,454)(91,424,150,455)(92,425,151,456)(93,426,152,457)(249,357,471,295)(250,358,472,296)(251,359,473,297)(252,360,474,298)(253,361,475,299)(254,362,476,300)(255,363,477,301)(256,364,478,302)(257,365,479,303)(258,366,480,304)(259,367,481,305)(260,368,482,306)(261,369,483,307)(262,370,484,308)(263,371,485,309)(264,372,486,310)(265,342,487,280)(266,343,488,281)(267,344,489,282)(268,345,490,283)(269,346,491,284)(270,347,492,285)(271,348,493,286)(272,349,494,287)(273,350,495,288)(274,351,496,289)(275,352,466,290)(276,353,467,291)(277,354,468,292)(278,355,469,293)(279,356,470,294)>;
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)(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465)(466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,182,360,87)(2,183,361,88)(3,184,362,89)(4,185,363,90)(5,186,364,91)(6,156,365,92)(7,157,366,93)(8,158,367,63)(9,159,368,64)(10,160,369,65)(11,161,370,66)(12,162,371,67)(13,163,372,68)(14,164,342,69)(15,165,343,70)(16,166,344,71)(17,167,345,72)(18,168,346,73)(19,169,347,74)(20,170,348,75)(21,171,349,76)(22,172,350,77)(23,173,351,78)(24,174,352,79)(25,175,353,80)(26,176,354,81)(27,177,355,82)(28,178,356,83)(29,179,357,84)(30,180,358,85)(31,181,359,86)(32,300,148,229)(33,301,149,230)(34,302,150,231)(35,303,151,232)(36,304,152,233)(37,305,153,234)(38,306,154,235)(39,307,155,236)(40,308,125,237)(41,309,126,238)(42,310,127,239)(43,280,128,240)(44,281,129,241)(45,282,130,242)(46,283,131,243)(47,284,132,244)(48,285,133,245)(49,286,134,246)(50,287,135,247)(51,288,136,248)(52,289,137,218)(53,290,138,219)(54,291,139,220)(55,292,140,221)(56,293,141,222)(57,294,142,223)(58,295,143,224)(59,296,144,225)(60,297,145,226)(61,298,146,227)(62,299,147,228)(94,396,465,266)(95,397,435,267)(96,398,436,268)(97,399,437,269)(98,400,438,270)(99,401,439,271)(100,402,440,272)(101,403,441,273)(102,373,442,274)(103,374,443,275)(104,375,444,276)(105,376,445,277)(106,377,446,278)(107,378,447,279)(108,379,448,249)(109,380,449,250)(110,381,450,251)(111,382,451,252)(112,383,452,253)(113,384,453,254)(114,385,454,255)(115,386,455,256)(116,387,456,257)(117,388,457,258)(118,389,458,259)(119,390,459,260)(120,391,460,261)(121,392,461,262)(122,393,462,263)(123,394,463,264)(124,395,464,265)(187,410,495,341)(188,411,496,311)(189,412,466,312)(190,413,467,313)(191,414,468,314)(192,415,469,315)(193,416,470,316)(194,417,471,317)(195,418,472,318)(196,419,473,319)(197,420,474,320)(198,421,475,321)(199,422,476,322)(200,423,477,323)(201,424,478,324)(202,425,479,325)(203,426,480,326)(204,427,481,327)(205,428,482,328)(206,429,483,329)(207,430,484,330)(208,431,485,331)(209,432,486,332)(210,433,487,333)(211,434,488,334)(212,404,489,335)(213,405,490,336)(214,406,491,337)(215,407,492,338)(216,408,493,339)(217,409,494,340), (1,197,227,382)(2,198,228,383)(3,199,229,384)(4,200,230,385)(5,201,231,386)(6,202,232,387)(7,203,233,388)(8,204,234,389)(9,205,235,390)(10,206,236,391)(11,207,237,392)(12,208,238,393)(13,209,239,394)(14,210,240,395)(15,211,241,396)(16,212,242,397)(17,213,243,398)(18,214,244,399)(19,215,245,400)(20,216,246,401)(21,217,247,402)(22,187,248,403)(23,188,218,373)(24,189,219,374)(25,190,220,375)(26,191,221,376)(27,192,222,377)(28,193,223,378)(29,194,224,379)(30,195,225,380)(31,196,226,381)(32,113,184,322)(33,114,185,323)(34,115,186,324)(35,116,156,325)(36,117,157,326)(37,118,158,327)(38,119,159,328)(39,120,160,329)(40,121,161,330)(41,122,162,331)(42,123,163,332)(43,124,164,333)(44,94,165,334)(45,95,166,335)(46,96,167,336)(47,97,168,337)(48,98,169,338)(49,99,170,339)(50,100,171,340)(51,101,172,341)(52,102,173,311)(53,103,174,312)(54,104,175,313)(55,105,176,314)(56,106,177,315)(57,107,178,316)(58,108,179,317)(59,109,180,318)(60,110,181,319)(61,111,182,320)(62,112,183,321)(63,427,153,458)(64,428,154,459)(65,429,155,460)(66,430,125,461)(67,431,126,462)(68,432,127,463)(69,433,128,464)(70,434,129,465)(71,404,130,435)(72,405,131,436)(73,406,132,437)(74,407,133,438)(75,408,134,439)(76,409,135,440)(77,410,136,441)(78,411,137,442)(79,412,138,443)(80,413,139,444)(81,414,140,445)(82,415,141,446)(83,416,142,447)(84,417,143,448)(85,418,144,449)(86,419,145,450)(87,420,146,451)(88,421,147,452)(89,422,148,453)(90,423,149,454)(91,424,150,455)(92,425,151,456)(93,426,152,457)(249,357,471,295)(250,358,472,296)(251,359,473,297)(252,360,474,298)(253,361,475,299)(254,362,476,300)(255,363,477,301)(256,364,478,302)(257,365,479,303)(258,366,480,304)(259,367,481,305)(260,368,482,306)(261,369,483,307)(262,370,484,308)(263,371,485,309)(264,372,486,310)(265,342,487,280)(266,343,488,281)(267,344,489,282)(268,345,490,283)(269,346,491,284)(270,347,492,285)(271,348,493,286)(272,349,494,287)(273,350,495,288)(274,351,496,289)(275,352,466,290)(276,353,467,291)(277,354,468,292)(278,355,469,293)(279,356,470,294) );
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),(435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465),(466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)], [(1,182,360,87),(2,183,361,88),(3,184,362,89),(4,185,363,90),(5,186,364,91),(6,156,365,92),(7,157,366,93),(8,158,367,63),(9,159,368,64),(10,160,369,65),(11,161,370,66),(12,162,371,67),(13,163,372,68),(14,164,342,69),(15,165,343,70),(16,166,344,71),(17,167,345,72),(18,168,346,73),(19,169,347,74),(20,170,348,75),(21,171,349,76),(22,172,350,77),(23,173,351,78),(24,174,352,79),(25,175,353,80),(26,176,354,81),(27,177,355,82),(28,178,356,83),(29,179,357,84),(30,180,358,85),(31,181,359,86),(32,300,148,229),(33,301,149,230),(34,302,150,231),(35,303,151,232),(36,304,152,233),(37,305,153,234),(38,306,154,235),(39,307,155,236),(40,308,125,237),(41,309,126,238),(42,310,127,239),(43,280,128,240),(44,281,129,241),(45,282,130,242),(46,283,131,243),(47,284,132,244),(48,285,133,245),(49,286,134,246),(50,287,135,247),(51,288,136,248),(52,289,137,218),(53,290,138,219),(54,291,139,220),(55,292,140,221),(56,293,141,222),(57,294,142,223),(58,295,143,224),(59,296,144,225),(60,297,145,226),(61,298,146,227),(62,299,147,228),(94,396,465,266),(95,397,435,267),(96,398,436,268),(97,399,437,269),(98,400,438,270),(99,401,439,271),(100,402,440,272),(101,403,441,273),(102,373,442,274),(103,374,443,275),(104,375,444,276),(105,376,445,277),(106,377,446,278),(107,378,447,279),(108,379,448,249),(109,380,449,250),(110,381,450,251),(111,382,451,252),(112,383,452,253),(113,384,453,254),(114,385,454,255),(115,386,455,256),(116,387,456,257),(117,388,457,258),(118,389,458,259),(119,390,459,260),(120,391,460,261),(121,392,461,262),(122,393,462,263),(123,394,463,264),(124,395,464,265),(187,410,495,341),(188,411,496,311),(189,412,466,312),(190,413,467,313),(191,414,468,314),(192,415,469,315),(193,416,470,316),(194,417,471,317),(195,418,472,318),(196,419,473,319),(197,420,474,320),(198,421,475,321),(199,422,476,322),(200,423,477,323),(201,424,478,324),(202,425,479,325),(203,426,480,326),(204,427,481,327),(205,428,482,328),(206,429,483,329),(207,430,484,330),(208,431,485,331),(209,432,486,332),(210,433,487,333),(211,434,488,334),(212,404,489,335),(213,405,490,336),(214,406,491,337),(215,407,492,338),(216,408,493,339),(217,409,494,340)], [(1,197,227,382),(2,198,228,383),(3,199,229,384),(4,200,230,385),(5,201,231,386),(6,202,232,387),(7,203,233,388),(8,204,234,389),(9,205,235,390),(10,206,236,391),(11,207,237,392),(12,208,238,393),(13,209,239,394),(14,210,240,395),(15,211,241,396),(16,212,242,397),(17,213,243,398),(18,214,244,399),(19,215,245,400),(20,216,246,401),(21,217,247,402),(22,187,248,403),(23,188,218,373),(24,189,219,374),(25,190,220,375),(26,191,221,376),(27,192,222,377),(28,193,223,378),(29,194,224,379),(30,195,225,380),(31,196,226,381),(32,113,184,322),(33,114,185,323),(34,115,186,324),(35,116,156,325),(36,117,157,326),(37,118,158,327),(38,119,159,328),(39,120,160,329),(40,121,161,330),(41,122,162,331),(42,123,163,332),(43,124,164,333),(44,94,165,334),(45,95,166,335),(46,96,167,336),(47,97,168,337),(48,98,169,338),(49,99,170,339),(50,100,171,340),(51,101,172,341),(52,102,173,311),(53,103,174,312),(54,104,175,313),(55,105,176,314),(56,106,177,315),(57,107,178,316),(58,108,179,317),(59,109,180,318),(60,110,181,319),(61,111,182,320),(62,112,183,321),(63,427,153,458),(64,428,154,459),(65,429,155,460),(66,430,125,461),(67,431,126,462),(68,432,127,463),(69,433,128,464),(70,434,129,465),(71,404,130,435),(72,405,131,436),(73,406,132,437),(74,407,133,438),(75,408,134,439),(76,409,135,440),(77,410,136,441),(78,411,137,442),(79,412,138,443),(80,413,139,444),(81,414,140,445),(82,415,141,446),(83,416,142,447),(84,417,143,448),(85,418,144,449),(86,419,145,450),(87,420,146,451),(88,421,147,452),(89,422,148,453),(90,423,149,454),(91,424,150,455),(92,425,151,456),(93,426,152,457),(249,357,471,295),(250,358,472,296),(251,359,473,297),(252,360,474,298),(253,361,475,299),(254,362,476,300),(255,363,477,301),(256,364,478,302),(257,365,479,303),(258,366,480,304),(259,367,481,305),(260,368,482,306),(261,369,483,307),(262,370,484,308),(263,371,485,309),(264,372,486,310),(265,342,487,280),(266,343,488,281),(267,344,489,282),(268,345,490,283),(269,346,491,284),(270,347,492,285),(271,348,493,286),(272,349,494,287),(273,350,495,288),(274,351,496,289),(275,352,466,290),(276,353,467,291),(277,354,468,292),(278,355,469,293),(279,356,470,294)]])
310 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 31A | ··· | 31AD | 62A | ··· | 62CL | 124A | ··· | 124FX |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
310 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C31 | C62 | C124 | D4 | Q8 | D4×C31 | Q8×C31 |
| kernel | C4⋊C4×C31 | C2×C124 | C124 | C4⋊C4 | C2×C4 | C4 | C62 | C62 | C2 | C2 |
| # reps | 1 | 3 | 4 | 30 | 90 | 120 | 1 | 1 | 30 | 30 |
Matrix representation of C4⋊C4×C31 ►in GL3(𝔽373) generated by
| 1 | 0 | 0 |
| 0 | 236 | 0 |
| 0 | 0 | 236 |
| 1 | 0 | 0 |
| 0 | 1 | 2 |
| 0 | 372 | 372 |
| 104 | 0 | 0 |
| 0 | 31 | 359 |
| 0 | 335 | 342 |
G:=sub<GL(3,GF(373))| [1,0,0,0,236,0,0,0,236],[1,0,0,0,1,372,0,2,372],[104,0,0,0,31,335,0,359,342] >;
C4⋊C4×C31 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{31} % in TeX
G:=Group("C4:C4xC31"); // GroupNames label
G:=SmallGroup(496,21);
// by ID
G=gap.SmallGroup(496,21);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,1240,1261,626]);
// Polycyclic
G:=Group<a,b,c|a^31=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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