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G = Dic107order 428 = 22·107

Dicyclic group

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

Aliases: Dic107, C107⋊C4, C214.C2, C2.D107, SmallGroup(428,1)

Series: Derived Chief Lower central Upper central

C1C107 — Dic107
C1C107C214 — Dic107
C107 — Dic107
C1C2

Generators and relations for Dic107
 G = < a,b | a214=1, b2=a107, bab-1=a-1 >

107C4

Smallest permutation representation of Dic107
Regular action on 428 points
Generators in S428
(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)
(1 261 108 368)(2 260 109 367)(3 259 110 366)(4 258 111 365)(5 257 112 364)(6 256 113 363)(7 255 114 362)(8 254 115 361)(9 253 116 360)(10 252 117 359)(11 251 118 358)(12 250 119 357)(13 249 120 356)(14 248 121 355)(15 247 122 354)(16 246 123 353)(17 245 124 352)(18 244 125 351)(19 243 126 350)(20 242 127 349)(21 241 128 348)(22 240 129 347)(23 239 130 346)(24 238 131 345)(25 237 132 344)(26 236 133 343)(27 235 134 342)(28 234 135 341)(29 233 136 340)(30 232 137 339)(31 231 138 338)(32 230 139 337)(33 229 140 336)(34 228 141 335)(35 227 142 334)(36 226 143 333)(37 225 144 332)(38 224 145 331)(39 223 146 330)(40 222 147 329)(41 221 148 328)(42 220 149 327)(43 219 150 326)(44 218 151 325)(45 217 152 324)(46 216 153 323)(47 215 154 322)(48 428 155 321)(49 427 156 320)(50 426 157 319)(51 425 158 318)(52 424 159 317)(53 423 160 316)(54 422 161 315)(55 421 162 314)(56 420 163 313)(57 419 164 312)(58 418 165 311)(59 417 166 310)(60 416 167 309)(61 415 168 308)(62 414 169 307)(63 413 170 306)(64 412 171 305)(65 411 172 304)(66 410 173 303)(67 409 174 302)(68 408 175 301)(69 407 176 300)(70 406 177 299)(71 405 178 298)(72 404 179 297)(73 403 180 296)(74 402 181 295)(75 401 182 294)(76 400 183 293)(77 399 184 292)(78 398 185 291)(79 397 186 290)(80 396 187 289)(81 395 188 288)(82 394 189 287)(83 393 190 286)(84 392 191 285)(85 391 192 284)(86 390 193 283)(87 389 194 282)(88 388 195 281)(89 387 196 280)(90 386 197 279)(91 385 198 278)(92 384 199 277)(93 383 200 276)(94 382 201 275)(95 381 202 274)(96 380 203 273)(97 379 204 272)(98 378 205 271)(99 377 206 270)(100 376 207 269)(101 375 208 268)(102 374 209 267)(103 373 210 266)(104 372 211 265)(105 371 212 264)(106 370 213 263)(107 369 214 262)

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

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

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

110 conjugacy classes

class 1  2 4A4B107A···107BA214A···214BA
order1244107···107214···214
size111071072···22···2

110 irreducible representations

dim11122
type+++-
imageC1C2C4D107Dic107
kernelDic107C214C107C2C1
# reps1125353

Matrix representation of Dic107 in GL3(𝔽857) generated by

85600
0615856
010
,
65000
094150
0541763
G:=sub<GL(3,GF(857))| [856,0,0,0,615,1,0,856,0],[650,0,0,0,94,541,0,150,763] >;

Dic107 in GAP, Magma, Sage, TeX

{\rm Dic}_{107}
% in TeX

G:=Group("Dic107");
// GroupNames label

G:=SmallGroup(428,1);
// by ID

G=gap.SmallGroup(428,1);
# by ID

G:=PCGroup([3,-2,-2,-107,6,3818]);
// Polycyclic

G:=Group<a,b|a^214=1,b^2=a^107,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic107 in TeX

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