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G = Dic114order 456 = 23·3·19

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic114, C4.D57, C572Q8, C76.1S3, C38.8D6, C6.8D38, C192Dic6, C32Dic38, C228.1C2, C12.1D19, C2.3D114, C114.8C22, Dic57.1C2, SmallGroup(456,34)

Series: Derived Chief Lower central Upper central

C1C114 — Dic114
C1C19C57C114Dic57 — Dic114
C57C114 — Dic114
C1C2C4

Generators and relations for Dic114
 G = < a,b | a228=1, b2=a114, bab-1=a-1 >

57C4
57C4
57Q8
19Dic3
19Dic3
3Dic19
3Dic19
19Dic6
3Dic38

Smallest permutation representation of Dic114
Regular action on 456 points
Generators in S456
(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)
(1 305 115 419)(2 304 116 418)(3 303 117 417)(4 302 118 416)(5 301 119 415)(6 300 120 414)(7 299 121 413)(8 298 122 412)(9 297 123 411)(10 296 124 410)(11 295 125 409)(12 294 126 408)(13 293 127 407)(14 292 128 406)(15 291 129 405)(16 290 130 404)(17 289 131 403)(18 288 132 402)(19 287 133 401)(20 286 134 400)(21 285 135 399)(22 284 136 398)(23 283 137 397)(24 282 138 396)(25 281 139 395)(26 280 140 394)(27 279 141 393)(28 278 142 392)(29 277 143 391)(30 276 144 390)(31 275 145 389)(32 274 146 388)(33 273 147 387)(34 272 148 386)(35 271 149 385)(36 270 150 384)(37 269 151 383)(38 268 152 382)(39 267 153 381)(40 266 154 380)(41 265 155 379)(42 264 156 378)(43 263 157 377)(44 262 158 376)(45 261 159 375)(46 260 160 374)(47 259 161 373)(48 258 162 372)(49 257 163 371)(50 256 164 370)(51 255 165 369)(52 254 166 368)(53 253 167 367)(54 252 168 366)(55 251 169 365)(56 250 170 364)(57 249 171 363)(58 248 172 362)(59 247 173 361)(60 246 174 360)(61 245 175 359)(62 244 176 358)(63 243 177 357)(64 242 178 356)(65 241 179 355)(66 240 180 354)(67 239 181 353)(68 238 182 352)(69 237 183 351)(70 236 184 350)(71 235 185 349)(72 234 186 348)(73 233 187 347)(74 232 188 346)(75 231 189 345)(76 230 190 344)(77 229 191 343)(78 456 192 342)(79 455 193 341)(80 454 194 340)(81 453 195 339)(82 452 196 338)(83 451 197 337)(84 450 198 336)(85 449 199 335)(86 448 200 334)(87 447 201 333)(88 446 202 332)(89 445 203 331)(90 444 204 330)(91 443 205 329)(92 442 206 328)(93 441 207 327)(94 440 208 326)(95 439 209 325)(96 438 210 324)(97 437 211 323)(98 436 212 322)(99 435 213 321)(100 434 214 320)(101 433 215 319)(102 432 216 318)(103 431 217 317)(104 430 218 316)(105 429 219 315)(106 428 220 314)(107 427 221 313)(108 426 222 312)(109 425 223 311)(110 424 224 310)(111 423 225 309)(112 422 226 308)(113 421 227 307)(114 420 228 306)

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

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

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

117 conjugacy classes

class 1  2  3 4A4B4C 6 12A12B19A···19I38A···38I57A···57R76A···76R114A···114R228A···228AJ
order1234446121219···1938···3857···5776···76114···114228···228
size11221141142222···22···22···22···22···22···2

117 irreducible representations

dim1112222222222
type++++-+-+++-+-
imageC1C2C2S3Q8D6Dic6D19D38D57Dic38D114Dic114
kernelDic114Dic57C228C76C57C38C19C12C6C4C3C2C1
# reps12111129918181836

Matrix representation of Dic114 in GL4(𝔽229) generated by

11110600
123400
0016523
0016267
,
4510000
19518400
0048161
00145181
G:=sub<GL(4,GF(229))| [111,123,0,0,106,4,0,0,0,0,165,162,0,0,23,67],[45,195,0,0,100,184,0,0,0,0,48,145,0,0,161,181] >;

Dic114 in GAP, Magma, Sage, TeX

{\rm Dic}_{114}
% in TeX

G:=Group("Dic114");
// GroupNames label

G:=SmallGroup(456,34);
// by ID

G=gap.SmallGroup(456,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-19,20,61,26,323,10804]);
// Polycyclic

G:=Group<a,b|a^228=1,b^2=a^114,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic114 in TeX

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