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G = Dic113order 452 = 22·113

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic113, C1132C4, C226.C2, C2.D113, SmallGroup(452,1)

Series: Derived Chief Lower central Upper central

C1C113 — Dic113
C1C113C226 — Dic113
C113 — Dic113
C1C2

Generators and relations for Dic113
 G = < a,b | a226=1, b2=a113, bab-1=a-1 >

113C4

Smallest permutation representation of Dic113
Regular action on 452 points
Generators in S452
(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)
(1 385 114 272)(2 384 115 271)(3 383 116 270)(4 382 117 269)(5 381 118 268)(6 380 119 267)(7 379 120 266)(8 378 121 265)(9 377 122 264)(10 376 123 263)(11 375 124 262)(12 374 125 261)(13 373 126 260)(14 372 127 259)(15 371 128 258)(16 370 129 257)(17 369 130 256)(18 368 131 255)(19 367 132 254)(20 366 133 253)(21 365 134 252)(22 364 135 251)(23 363 136 250)(24 362 137 249)(25 361 138 248)(26 360 139 247)(27 359 140 246)(28 358 141 245)(29 357 142 244)(30 356 143 243)(31 355 144 242)(32 354 145 241)(33 353 146 240)(34 352 147 239)(35 351 148 238)(36 350 149 237)(37 349 150 236)(38 348 151 235)(39 347 152 234)(40 346 153 233)(41 345 154 232)(42 344 155 231)(43 343 156 230)(44 342 157 229)(45 341 158 228)(46 340 159 227)(47 339 160 452)(48 338 161 451)(49 337 162 450)(50 336 163 449)(51 335 164 448)(52 334 165 447)(53 333 166 446)(54 332 167 445)(55 331 168 444)(56 330 169 443)(57 329 170 442)(58 328 171 441)(59 327 172 440)(60 326 173 439)(61 325 174 438)(62 324 175 437)(63 323 176 436)(64 322 177 435)(65 321 178 434)(66 320 179 433)(67 319 180 432)(68 318 181 431)(69 317 182 430)(70 316 183 429)(71 315 184 428)(72 314 185 427)(73 313 186 426)(74 312 187 425)(75 311 188 424)(76 310 189 423)(77 309 190 422)(78 308 191 421)(79 307 192 420)(80 306 193 419)(81 305 194 418)(82 304 195 417)(83 303 196 416)(84 302 197 415)(85 301 198 414)(86 300 199 413)(87 299 200 412)(88 298 201 411)(89 297 202 410)(90 296 203 409)(91 295 204 408)(92 294 205 407)(93 293 206 406)(94 292 207 405)(95 291 208 404)(96 290 209 403)(97 289 210 402)(98 288 211 401)(99 287 212 400)(100 286 213 399)(101 285 214 398)(102 284 215 397)(103 283 216 396)(104 282 217 395)(105 281 218 394)(106 280 219 393)(107 279 220 392)(108 278 221 391)(109 277 222 390)(110 276 223 389)(111 275 224 388)(112 274 225 387)(113 273 226 386)

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

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

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

116 conjugacy classes

class 1  2 4A4B113A···113BD226A···226BD
order1244113···113226···226
size111131132···22···2

116 irreducible representations

dim11122
type+++-
imageC1C2C4D113Dic113
kernelDic113C226C113C2C1
# reps1125656

Matrix representation of Dic113 in GL2(𝔽2713) generated by

6121
27120
,
2028582
2000685
G:=sub<GL(2,GF(2713))| [612,2712,1,0],[2028,2000,582,685] >;

Dic113 in GAP, Magma, Sage, TeX

{\rm Dic}_{113}
% in TeX

G:=Group("Dic113");
// GroupNames label

G:=SmallGroup(452,1);
// by ID

G=gap.SmallGroup(452,1);
# by ID

G:=PCGroup([3,-2,-2,-113,6,4034]);
// Polycyclic

G:=Group<a,b|a^226=1,b^2=a^113,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic113 in TeX

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