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G = Dic111order 444 = 22·3·37

Dicyclic group

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

Aliases: Dic111, C74.S3, C6.D37, C3⋊Dic37, C1113C4, C2.D111, C372Dic3, C222.1C2, SmallGroup(444,5)

Series: Derived Chief Lower central Upper central

C1C111 — Dic111
C1C37C111C222 — Dic111
C111 — Dic111
C1C2

Generators and relations for Dic111
 G = < a,b | a222=1, b2=a111, bab-1=a-1 >

111C4
37Dic3
3Dic37

Smallest permutation representation of Dic111
Regular action on 444 points
Generators in S444
(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)
(1 437 112 326)(2 436 113 325)(3 435 114 324)(4 434 115 323)(5 433 116 322)(6 432 117 321)(7 431 118 320)(8 430 119 319)(9 429 120 318)(10 428 121 317)(11 427 122 316)(12 426 123 315)(13 425 124 314)(14 424 125 313)(15 423 126 312)(16 422 127 311)(17 421 128 310)(18 420 129 309)(19 419 130 308)(20 418 131 307)(21 417 132 306)(22 416 133 305)(23 415 134 304)(24 414 135 303)(25 413 136 302)(26 412 137 301)(27 411 138 300)(28 410 139 299)(29 409 140 298)(30 408 141 297)(31 407 142 296)(32 406 143 295)(33 405 144 294)(34 404 145 293)(35 403 146 292)(36 402 147 291)(37 401 148 290)(38 400 149 289)(39 399 150 288)(40 398 151 287)(41 397 152 286)(42 396 153 285)(43 395 154 284)(44 394 155 283)(45 393 156 282)(46 392 157 281)(47 391 158 280)(48 390 159 279)(49 389 160 278)(50 388 161 277)(51 387 162 276)(52 386 163 275)(53 385 164 274)(54 384 165 273)(55 383 166 272)(56 382 167 271)(57 381 168 270)(58 380 169 269)(59 379 170 268)(60 378 171 267)(61 377 172 266)(62 376 173 265)(63 375 174 264)(64 374 175 263)(65 373 176 262)(66 372 177 261)(67 371 178 260)(68 370 179 259)(69 369 180 258)(70 368 181 257)(71 367 182 256)(72 366 183 255)(73 365 184 254)(74 364 185 253)(75 363 186 252)(76 362 187 251)(77 361 188 250)(78 360 189 249)(79 359 190 248)(80 358 191 247)(81 357 192 246)(82 356 193 245)(83 355 194 244)(84 354 195 243)(85 353 196 242)(86 352 197 241)(87 351 198 240)(88 350 199 239)(89 349 200 238)(90 348 201 237)(91 347 202 236)(92 346 203 235)(93 345 204 234)(94 344 205 233)(95 343 206 232)(96 342 207 231)(97 341 208 230)(98 340 209 229)(99 339 210 228)(100 338 211 227)(101 337 212 226)(102 336 213 225)(103 335 214 224)(104 334 215 223)(105 333 216 444)(106 332 217 443)(107 331 218 442)(108 330 219 441)(109 329 220 440)(110 328 221 439)(111 327 222 438)

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

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

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

114 conjugacy classes

class 1  2  3 4A4B 6 37A···37R74A···74R111A···111AJ222A···222AJ
order12344637···3774···74111···111222···222
size11211111122···22···22···22···2

114 irreducible representations

dim111222222
type+++-+-+-
imageC1C2C4S3Dic3D37Dic37D111Dic111
kernelDic111C222C111C74C37C6C3C2C1
# reps1121118183636

Matrix representation of Dic111 in GL2(𝔽1777) generated by

9831316
4611700
,
102959
15791675
G:=sub<GL(2,GF(1777))| [983,461,1316,1700],[102,1579,959,1675] >;

Dic111 in GAP, Magma, Sage, TeX

{\rm Dic}_{111}
% in TeX

G:=Group("Dic111");
// GroupNames label

G:=SmallGroup(444,5);
// by ID

G=gap.SmallGroup(444,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-37,8,98,6915]);
// Polycyclic

G:=Group<a,b|a^222=1,b^2=a^111,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic111 in TeX

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