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G = Dic106order 424 = 23·53

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic106, C53⋊Q8, C4.D53, C212.1C2, C2.3D106, C106.1C22, Dic53.1C2, SmallGroup(424,4)

Series: Derived Chief Lower central Upper central

C1C106 — Dic106
C1C53C106Dic53 — Dic106
C53C106 — Dic106
C1C2C4

Generators and relations for Dic106
 G = < a,b | a212=1, b2=a106, bab-1=a-1 >

53C4
53C4
53Q8

Smallest permutation representation of Dic106
Regular action on 424 points
Generators in S424
(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)
(1 308 107 414)(2 307 108 413)(3 306 109 412)(4 305 110 411)(5 304 111 410)(6 303 112 409)(7 302 113 408)(8 301 114 407)(9 300 115 406)(10 299 116 405)(11 298 117 404)(12 297 118 403)(13 296 119 402)(14 295 120 401)(15 294 121 400)(16 293 122 399)(17 292 123 398)(18 291 124 397)(19 290 125 396)(20 289 126 395)(21 288 127 394)(22 287 128 393)(23 286 129 392)(24 285 130 391)(25 284 131 390)(26 283 132 389)(27 282 133 388)(28 281 134 387)(29 280 135 386)(30 279 136 385)(31 278 137 384)(32 277 138 383)(33 276 139 382)(34 275 140 381)(35 274 141 380)(36 273 142 379)(37 272 143 378)(38 271 144 377)(39 270 145 376)(40 269 146 375)(41 268 147 374)(42 267 148 373)(43 266 149 372)(44 265 150 371)(45 264 151 370)(46 263 152 369)(47 262 153 368)(48 261 154 367)(49 260 155 366)(50 259 156 365)(51 258 157 364)(52 257 158 363)(53 256 159 362)(54 255 160 361)(55 254 161 360)(56 253 162 359)(57 252 163 358)(58 251 164 357)(59 250 165 356)(60 249 166 355)(61 248 167 354)(62 247 168 353)(63 246 169 352)(64 245 170 351)(65 244 171 350)(66 243 172 349)(67 242 173 348)(68 241 174 347)(69 240 175 346)(70 239 176 345)(71 238 177 344)(72 237 178 343)(73 236 179 342)(74 235 180 341)(75 234 181 340)(76 233 182 339)(77 232 183 338)(78 231 184 337)(79 230 185 336)(80 229 186 335)(81 228 187 334)(82 227 188 333)(83 226 189 332)(84 225 190 331)(85 224 191 330)(86 223 192 329)(87 222 193 328)(88 221 194 327)(89 220 195 326)(90 219 196 325)(91 218 197 324)(92 217 198 323)(93 216 199 322)(94 215 200 321)(95 214 201 320)(96 213 202 319)(97 424 203 318)(98 423 204 317)(99 422 205 316)(100 421 206 315)(101 420 207 314)(102 419 208 313)(103 418 209 312)(104 417 210 311)(105 416 211 310)(106 415 212 309)

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

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

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

109 conjugacy classes

class 1  2 4A4B4C53A···53Z106A···106Z212A···212AZ
order1244453···53106···106212···212
size1121061062···22···22···2

109 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D53D106Dic106
kernelDic106Dic53C212C53C4C2C1
# reps1211262652

Matrix representation of Dic106 in GL2(𝔽1061) generated by

324185
876641
,
768441
977293
G:=sub<GL(2,GF(1061))| [324,876,185,641],[768,977,441,293] >;

Dic106 in GAP, Magma, Sage, TeX

{\rm Dic}_{106}
% in TeX

G:=Group("Dic106");
// GroupNames label

G:=SmallGroup(424,4);
// by ID

G=gap.SmallGroup(424,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-53,16,49,21,6659]);
// Polycyclic

G:=Group<a,b|a^212=1,b^2=a^106,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic106 in TeX

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