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G = C4×C116order 464 = 24·29

Abelian group of type [4,116]

direct product, abelian, monomial, 2-elementary

Aliases: C4×C116, SmallGroup(464,20)

Series: Derived Chief Lower central Upper central

C1 — C4×C116
C1C2C22C2×C58C2×C116 — C4×C116
C1 — C4×C116
C1 — C4×C116

Generators and relations for C4×C116
 G = < a,b | a4=b116=1, ab=ba >


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

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

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

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

464 conjugacy classes

class 1 2A2B2C4A···4L29A···29AB58A···58CF116A···116LX
order12224···429···2958···58116···116
size11111···11···11···11···1

464 irreducible representations

dim111111
type++
imageC1C2C4C29C58C116
kernelC4×C116C2×C116C116C42C2×C4C4
# reps13122884336

Matrix representation of C4×C116 in GL2(𝔽233) generated by

2320
089
,
2190
030
G:=sub<GL(2,GF(233))| [232,0,0,89],[219,0,0,30] >;

C4×C116 in GAP, Magma, Sage, TeX

C_4\times C_{116}
% in TeX

G:=Group("C4xC116");
// GroupNames label

G:=SmallGroup(464,20);
// by ID

G=gap.SmallGroup(464,20);
# by ID

G:=PCGroup([5,-2,-2,-29,-2,-2,580,1166]);
// Polycyclic

G:=Group<a,b|a^4=b^116=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C4×C116 in TeX

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