direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C4⋊C4×C29, C4⋊C116, C116⋊5C4, C58.3Q8, C58.13D4, C2.(Q8×C29), (C2×C4).1C58, C2.2(D4×C29), C58.18(C2×C4), C2.2(C2×C116), (C2×C116).2C2, C22.3(C2×C58), (C2×C58).14C22, SmallGroup(464,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4×C29
G = < a,b,c | a29=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4⋊C4×C29
►Regular action on 464 points
Generators in S464
(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)
(1 286 77 382)(2 287 78 383)(3 288 79 384)(4 289 80 385)(5 290 81 386)(6 262 82 387)(7 263 83 388)(8 264 84 389)(9 265 85 390)(10 266 86 391)(11 267 87 392)(12 268 59 393)(13 269 60 394)(14 270 61 395)(15 271 62 396)(16 272 63 397)(17 273 64 398)(18 274 65 399)(19 275 66 400)(20 276 67 401)(21 277 68 402)(22 278 69 403)(23 279 70 404)(24 280 71 405)(25 281 72 406)(26 282 73 378)(27 283 74 379)(28 284 75 380)(29 285 76 381)(30 411 360 225)(31 412 361 226)(32 413 362 227)(33 414 363 228)(34 415 364 229)(35 416 365 230)(36 417 366 231)(37 418 367 232)(38 419 368 204)(39 420 369 205)(40 421 370 206)(41 422 371 207)(42 423 372 208)(43 424 373 209)(44 425 374 210)(45 426 375 211)(46 427 376 212)(47 428 377 213)(48 429 349 214)(49 430 350 215)(50 431 351 216)(51 432 352 217)(52 433 353 218)(53 434 354 219)(54 435 355 220)(55 407 356 221)(56 408 357 222)(57 409 358 223)(58 410 359 224)(88 165 248 345)(89 166 249 346)(90 167 250 347)(91 168 251 348)(92 169 252 320)(93 170 253 321)(94 171 254 322)(95 172 255 323)(96 173 256 324)(97 174 257 325)(98 146 258 326)(99 147 259 327)(100 148 260 328)(101 149 261 329)(102 150 233 330)(103 151 234 331)(104 152 235 332)(105 153 236 333)(106 154 237 334)(107 155 238 335)(108 156 239 336)(109 157 240 337)(110 158 241 338)(111 159 242 339)(112 160 243 340)(113 161 244 341)(114 162 245 342)(115 163 246 343)(116 164 247 344)(117 202 310 449)(118 203 311 450)(119 175 312 451)(120 176 313 452)(121 177 314 453)(122 178 315 454)(123 179 316 455)(124 180 317 456)(125 181 318 457)(126 182 319 458)(127 183 291 459)(128 184 292 460)(129 185 293 461)(130 186 294 462)(131 187 295 463)(132 188 296 464)(133 189 297 436)(134 190 298 437)(135 191 299 438)(136 192 300 439)(137 193 301 440)(138 194 302 441)(139 195 303 442)(140 196 304 443)(141 197 305 444)(142 198 306 445)(143 199 307 446)(144 200 308 447)(145 201 309 448)
(1 369 158 137)(2 370 159 138)(3 371 160 139)(4 372 161 140)(5 373 162 141)(6 374 163 142)(7 375 164 143)(8 376 165 144)(9 377 166 145)(10 349 167 117)(11 350 168 118)(12 351 169 119)(13 352 170 120)(14 353 171 121)(15 354 172 122)(16 355 173 123)(17 356 174 124)(18 357 146 125)(19 358 147 126)(20 359 148 127)(21 360 149 128)(22 361 150 129)(23 362 151 130)(24 363 152 131)(25 364 153 132)(26 365 154 133)(27 366 155 134)(28 367 156 135)(29 368 157 136)(30 329 292 68)(31 330 293 69)(32 331 294 70)(33 332 295 71)(34 333 296 72)(35 334 297 73)(36 335 298 74)(37 336 299 75)(38 337 300 76)(39 338 301 77)(40 339 302 78)(41 340 303 79)(42 341 304 80)(43 342 305 81)(44 343 306 82)(45 344 307 83)(46 345 308 84)(47 346 309 85)(48 347 310 86)(49 348 311 87)(50 320 312 59)(51 321 313 60)(52 322 314 61)(53 323 315 62)(54 324 316 63)(55 325 317 64)(56 326 318 65)(57 327 319 66)(58 328 291 67)(88 200 389 212)(89 201 390 213)(90 202 391 214)(91 203 392 215)(92 175 393 216)(93 176 394 217)(94 177 395 218)(95 178 396 219)(96 179 397 220)(97 180 398 221)(98 181 399 222)(99 182 400 223)(100 183 401 224)(101 184 402 225)(102 185 403 226)(103 186 404 227)(104 187 405 228)(105 188 406 229)(106 189 378 230)(107 190 379 231)(108 191 380 232)(109 192 381 204)(110 193 382 205)(111 194 383 206)(112 195 384 207)(113 196 385 208)(114 197 386 209)(115 198 387 210)(116 199 388 211)(233 461 278 412)(234 462 279 413)(235 463 280 414)(236 464 281 415)(237 436 282 416)(238 437 283 417)(239 438 284 418)(240 439 285 419)(241 440 286 420)(242 441 287 421)(243 442 288 422)(244 443 289 423)(245 444 290 424)(246 445 262 425)(247 446 263 426)(248 447 264 427)(249 448 265 428)(250 449 266 429)(251 450 267 430)(252 451 268 431)(253 452 269 432)(254 453 270 433)(255 454 271 434)(256 455 272 435)(257 456 273 407)(258 457 274 408)(259 458 275 409)(260 459 276 410)(261 460 277 411)
G:=sub<Sym(464)| (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), (1,286,77,382)(2,287,78,383)(3,288,79,384)(4,289,80,385)(5,290,81,386)(6,262,82,387)(7,263,83,388)(8,264,84,389)(9,265,85,390)(10,266,86,391)(11,267,87,392)(12,268,59,393)(13,269,60,394)(14,270,61,395)(15,271,62,396)(16,272,63,397)(17,273,64,398)(18,274,65,399)(19,275,66,400)(20,276,67,401)(21,277,68,402)(22,278,69,403)(23,279,70,404)(24,280,71,405)(25,281,72,406)(26,282,73,378)(27,283,74,379)(28,284,75,380)(29,285,76,381)(30,411,360,225)(31,412,361,226)(32,413,362,227)(33,414,363,228)(34,415,364,229)(35,416,365,230)(36,417,366,231)(37,418,367,232)(38,419,368,204)(39,420,369,205)(40,421,370,206)(41,422,371,207)(42,423,372,208)(43,424,373,209)(44,425,374,210)(45,426,375,211)(46,427,376,212)(47,428,377,213)(48,429,349,214)(49,430,350,215)(50,431,351,216)(51,432,352,217)(52,433,353,218)(53,434,354,219)(54,435,355,220)(55,407,356,221)(56,408,357,222)(57,409,358,223)(58,410,359,224)(88,165,248,345)(89,166,249,346)(90,167,250,347)(91,168,251,348)(92,169,252,320)(93,170,253,321)(94,171,254,322)(95,172,255,323)(96,173,256,324)(97,174,257,325)(98,146,258,326)(99,147,259,327)(100,148,260,328)(101,149,261,329)(102,150,233,330)(103,151,234,331)(104,152,235,332)(105,153,236,333)(106,154,237,334)(107,155,238,335)(108,156,239,336)(109,157,240,337)(110,158,241,338)(111,159,242,339)(112,160,243,340)(113,161,244,341)(114,162,245,342)(115,163,246,343)(116,164,247,344)(117,202,310,449)(118,203,311,450)(119,175,312,451)(120,176,313,452)(121,177,314,453)(122,178,315,454)(123,179,316,455)(124,180,317,456)(125,181,318,457)(126,182,319,458)(127,183,291,459)(128,184,292,460)(129,185,293,461)(130,186,294,462)(131,187,295,463)(132,188,296,464)(133,189,297,436)(134,190,298,437)(135,191,299,438)(136,192,300,439)(137,193,301,440)(138,194,302,441)(139,195,303,442)(140,196,304,443)(141,197,305,444)(142,198,306,445)(143,199,307,446)(144,200,308,447)(145,201,309,448), (1,369,158,137)(2,370,159,138)(3,371,160,139)(4,372,161,140)(5,373,162,141)(6,374,163,142)(7,375,164,143)(8,376,165,144)(9,377,166,145)(10,349,167,117)(11,350,168,118)(12,351,169,119)(13,352,170,120)(14,353,171,121)(15,354,172,122)(16,355,173,123)(17,356,174,124)(18,357,146,125)(19,358,147,126)(20,359,148,127)(21,360,149,128)(22,361,150,129)(23,362,151,130)(24,363,152,131)(25,364,153,132)(26,365,154,133)(27,366,155,134)(28,367,156,135)(29,368,157,136)(30,329,292,68)(31,330,293,69)(32,331,294,70)(33,332,295,71)(34,333,296,72)(35,334,297,73)(36,335,298,74)(37,336,299,75)(38,337,300,76)(39,338,301,77)(40,339,302,78)(41,340,303,79)(42,341,304,80)(43,342,305,81)(44,343,306,82)(45,344,307,83)(46,345,308,84)(47,346,309,85)(48,347,310,86)(49,348,311,87)(50,320,312,59)(51,321,313,60)(52,322,314,61)(53,323,315,62)(54,324,316,63)(55,325,317,64)(56,326,318,65)(57,327,319,66)(58,328,291,67)(88,200,389,212)(89,201,390,213)(90,202,391,214)(91,203,392,215)(92,175,393,216)(93,176,394,217)(94,177,395,218)(95,178,396,219)(96,179,397,220)(97,180,398,221)(98,181,399,222)(99,182,400,223)(100,183,401,224)(101,184,402,225)(102,185,403,226)(103,186,404,227)(104,187,405,228)(105,188,406,229)(106,189,378,230)(107,190,379,231)(108,191,380,232)(109,192,381,204)(110,193,382,205)(111,194,383,206)(112,195,384,207)(113,196,385,208)(114,197,386,209)(115,198,387,210)(116,199,388,211)(233,461,278,412)(234,462,279,413)(235,463,280,414)(236,464,281,415)(237,436,282,416)(238,437,283,417)(239,438,284,418)(240,439,285,419)(241,440,286,420)(242,441,287,421)(243,442,288,422)(244,443,289,423)(245,444,290,424)(246,445,262,425)(247,446,263,426)(248,447,264,427)(249,448,265,428)(250,449,266,429)(251,450,267,430)(252,451,268,431)(253,452,269,432)(254,453,270,433)(255,454,271,434)(256,455,272,435)(257,456,273,407)(258,457,274,408)(259,458,275,409)(260,459,276,410)(261,460,277,411)>;
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), (1,286,77,382)(2,287,78,383)(3,288,79,384)(4,289,80,385)(5,290,81,386)(6,262,82,387)(7,263,83,388)(8,264,84,389)(9,265,85,390)(10,266,86,391)(11,267,87,392)(12,268,59,393)(13,269,60,394)(14,270,61,395)(15,271,62,396)(16,272,63,397)(17,273,64,398)(18,274,65,399)(19,275,66,400)(20,276,67,401)(21,277,68,402)(22,278,69,403)(23,279,70,404)(24,280,71,405)(25,281,72,406)(26,282,73,378)(27,283,74,379)(28,284,75,380)(29,285,76,381)(30,411,360,225)(31,412,361,226)(32,413,362,227)(33,414,363,228)(34,415,364,229)(35,416,365,230)(36,417,366,231)(37,418,367,232)(38,419,368,204)(39,420,369,205)(40,421,370,206)(41,422,371,207)(42,423,372,208)(43,424,373,209)(44,425,374,210)(45,426,375,211)(46,427,376,212)(47,428,377,213)(48,429,349,214)(49,430,350,215)(50,431,351,216)(51,432,352,217)(52,433,353,218)(53,434,354,219)(54,435,355,220)(55,407,356,221)(56,408,357,222)(57,409,358,223)(58,410,359,224)(88,165,248,345)(89,166,249,346)(90,167,250,347)(91,168,251,348)(92,169,252,320)(93,170,253,321)(94,171,254,322)(95,172,255,323)(96,173,256,324)(97,174,257,325)(98,146,258,326)(99,147,259,327)(100,148,260,328)(101,149,261,329)(102,150,233,330)(103,151,234,331)(104,152,235,332)(105,153,236,333)(106,154,237,334)(107,155,238,335)(108,156,239,336)(109,157,240,337)(110,158,241,338)(111,159,242,339)(112,160,243,340)(113,161,244,341)(114,162,245,342)(115,163,246,343)(116,164,247,344)(117,202,310,449)(118,203,311,450)(119,175,312,451)(120,176,313,452)(121,177,314,453)(122,178,315,454)(123,179,316,455)(124,180,317,456)(125,181,318,457)(126,182,319,458)(127,183,291,459)(128,184,292,460)(129,185,293,461)(130,186,294,462)(131,187,295,463)(132,188,296,464)(133,189,297,436)(134,190,298,437)(135,191,299,438)(136,192,300,439)(137,193,301,440)(138,194,302,441)(139,195,303,442)(140,196,304,443)(141,197,305,444)(142,198,306,445)(143,199,307,446)(144,200,308,447)(145,201,309,448), (1,369,158,137)(2,370,159,138)(3,371,160,139)(4,372,161,140)(5,373,162,141)(6,374,163,142)(7,375,164,143)(8,376,165,144)(9,377,166,145)(10,349,167,117)(11,350,168,118)(12,351,169,119)(13,352,170,120)(14,353,171,121)(15,354,172,122)(16,355,173,123)(17,356,174,124)(18,357,146,125)(19,358,147,126)(20,359,148,127)(21,360,149,128)(22,361,150,129)(23,362,151,130)(24,363,152,131)(25,364,153,132)(26,365,154,133)(27,366,155,134)(28,367,156,135)(29,368,157,136)(30,329,292,68)(31,330,293,69)(32,331,294,70)(33,332,295,71)(34,333,296,72)(35,334,297,73)(36,335,298,74)(37,336,299,75)(38,337,300,76)(39,338,301,77)(40,339,302,78)(41,340,303,79)(42,341,304,80)(43,342,305,81)(44,343,306,82)(45,344,307,83)(46,345,308,84)(47,346,309,85)(48,347,310,86)(49,348,311,87)(50,320,312,59)(51,321,313,60)(52,322,314,61)(53,323,315,62)(54,324,316,63)(55,325,317,64)(56,326,318,65)(57,327,319,66)(58,328,291,67)(88,200,389,212)(89,201,390,213)(90,202,391,214)(91,203,392,215)(92,175,393,216)(93,176,394,217)(94,177,395,218)(95,178,396,219)(96,179,397,220)(97,180,398,221)(98,181,399,222)(99,182,400,223)(100,183,401,224)(101,184,402,225)(102,185,403,226)(103,186,404,227)(104,187,405,228)(105,188,406,229)(106,189,378,230)(107,190,379,231)(108,191,380,232)(109,192,381,204)(110,193,382,205)(111,194,383,206)(112,195,384,207)(113,196,385,208)(114,197,386,209)(115,198,387,210)(116,199,388,211)(233,461,278,412)(234,462,279,413)(235,463,280,414)(236,464,281,415)(237,436,282,416)(238,437,283,417)(239,438,284,418)(240,439,285,419)(241,440,286,420)(242,441,287,421)(243,442,288,422)(244,443,289,423)(245,444,290,424)(246,445,262,425)(247,446,263,426)(248,447,264,427)(249,448,265,428)(250,449,266,429)(251,450,267,430)(252,451,268,431)(253,452,269,432)(254,453,270,433)(255,454,271,434)(256,455,272,435)(257,456,273,407)(258,457,274,408)(259,458,275,409)(260,459,276,410)(261,460,277,411) );
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)], [(1,286,77,382),(2,287,78,383),(3,288,79,384),(4,289,80,385),(5,290,81,386),(6,262,82,387),(7,263,83,388),(8,264,84,389),(9,265,85,390),(10,266,86,391),(11,267,87,392),(12,268,59,393),(13,269,60,394),(14,270,61,395),(15,271,62,396),(16,272,63,397),(17,273,64,398),(18,274,65,399),(19,275,66,400),(20,276,67,401),(21,277,68,402),(22,278,69,403),(23,279,70,404),(24,280,71,405),(25,281,72,406),(26,282,73,378),(27,283,74,379),(28,284,75,380),(29,285,76,381),(30,411,360,225),(31,412,361,226),(32,413,362,227),(33,414,363,228),(34,415,364,229),(35,416,365,230),(36,417,366,231),(37,418,367,232),(38,419,368,204),(39,420,369,205),(40,421,370,206),(41,422,371,207),(42,423,372,208),(43,424,373,209),(44,425,374,210),(45,426,375,211),(46,427,376,212),(47,428,377,213),(48,429,349,214),(49,430,350,215),(50,431,351,216),(51,432,352,217),(52,433,353,218),(53,434,354,219),(54,435,355,220),(55,407,356,221),(56,408,357,222),(57,409,358,223),(58,410,359,224),(88,165,248,345),(89,166,249,346),(90,167,250,347),(91,168,251,348),(92,169,252,320),(93,170,253,321),(94,171,254,322),(95,172,255,323),(96,173,256,324),(97,174,257,325),(98,146,258,326),(99,147,259,327),(100,148,260,328),(101,149,261,329),(102,150,233,330),(103,151,234,331),(104,152,235,332),(105,153,236,333),(106,154,237,334),(107,155,238,335),(108,156,239,336),(109,157,240,337),(110,158,241,338),(111,159,242,339),(112,160,243,340),(113,161,244,341),(114,162,245,342),(115,163,246,343),(116,164,247,344),(117,202,310,449),(118,203,311,450),(119,175,312,451),(120,176,313,452),(121,177,314,453),(122,178,315,454),(123,179,316,455),(124,180,317,456),(125,181,318,457),(126,182,319,458),(127,183,291,459),(128,184,292,460),(129,185,293,461),(130,186,294,462),(131,187,295,463),(132,188,296,464),(133,189,297,436),(134,190,298,437),(135,191,299,438),(136,192,300,439),(137,193,301,440),(138,194,302,441),(139,195,303,442),(140,196,304,443),(141,197,305,444),(142,198,306,445),(143,199,307,446),(144,200,308,447),(145,201,309,448)], [(1,369,158,137),(2,370,159,138),(3,371,160,139),(4,372,161,140),(5,373,162,141),(6,374,163,142),(7,375,164,143),(8,376,165,144),(9,377,166,145),(10,349,167,117),(11,350,168,118),(12,351,169,119),(13,352,170,120),(14,353,171,121),(15,354,172,122),(16,355,173,123),(17,356,174,124),(18,357,146,125),(19,358,147,126),(20,359,148,127),(21,360,149,128),(22,361,150,129),(23,362,151,130),(24,363,152,131),(25,364,153,132),(26,365,154,133),(27,366,155,134),(28,367,156,135),(29,368,157,136),(30,329,292,68),(31,330,293,69),(32,331,294,70),(33,332,295,71),(34,333,296,72),(35,334,297,73),(36,335,298,74),(37,336,299,75),(38,337,300,76),(39,338,301,77),(40,339,302,78),(41,340,303,79),(42,341,304,80),(43,342,305,81),(44,343,306,82),(45,344,307,83),(46,345,308,84),(47,346,309,85),(48,347,310,86),(49,348,311,87),(50,320,312,59),(51,321,313,60),(52,322,314,61),(53,323,315,62),(54,324,316,63),(55,325,317,64),(56,326,318,65),(57,327,319,66),(58,328,291,67),(88,200,389,212),(89,201,390,213),(90,202,391,214),(91,203,392,215),(92,175,393,216),(93,176,394,217),(94,177,395,218),(95,178,396,219),(96,179,397,220),(97,180,398,221),(98,181,399,222),(99,182,400,223),(100,183,401,224),(101,184,402,225),(102,185,403,226),(103,186,404,227),(104,187,405,228),(105,188,406,229),(106,189,378,230),(107,190,379,231),(108,191,380,232),(109,192,381,204),(110,193,382,205),(111,194,383,206),(112,195,384,207),(113,196,385,208),(114,197,386,209),(115,198,387,210),(116,199,388,211),(233,461,278,412),(234,462,279,413),(235,463,280,414),(236,464,281,415),(237,436,282,416),(238,437,283,417),(239,438,284,418),(240,439,285,419),(241,440,286,420),(242,441,287,421),(243,442,288,422),(244,443,289,423),(245,444,290,424),(246,445,262,425),(247,446,263,426),(248,447,264,427),(249,448,265,428),(250,449,266,429),(251,450,267,430),(252,451,268,431),(253,452,269,432),(254,453,270,433),(255,454,271,434),(256,455,272,435),(257,456,273,407),(258,457,274,408),(259,458,275,409),(260,459,276,410),(261,460,277,411)]])
290 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 29A | ··· | 29AB | 58A | ··· | 58CF | 116A | ··· | 116FL |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
290 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C29 | C58 | C116 | D4 | Q8 | D4×C29 | Q8×C29 |
| kernel | C4⋊C4×C29 | C2×C116 | C116 | C4⋊C4 | C2×C4 | C4 | C58 | C58 | C2 | C2 |
| # reps | 1 | 3 | 4 | 28 | 84 | 112 | 1 | 1 | 28 | 28 |
Matrix representation of C4⋊C4×C29 ►in GL3(𝔽233) generated by
| 1 | 0 | 0 |
| 0 | 92 | 0 |
| 0 | 0 | 92 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 232 | 0 |
| 144 | 0 | 0 |
| 0 | 135 | 154 |
| 0 | 154 | 98 |
G:=sub<GL(3,GF(233))| [1,0,0,0,92,0,0,0,92],[1,0,0,0,0,232,0,1,0],[144,0,0,0,135,154,0,154,98] >;
C4⋊C4×C29 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\times C_{29} % in TeX
G:=Group("C4:C4xC29"); // GroupNames label
G:=SmallGroup(464,22);
// by ID
G=gap.SmallGroup(464,22);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,1160,1181,586]);
// Polycyclic
G:=Group<a,b,c|a^29=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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