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## G = Dic105order 420 = 22·3·5·7

### Dicyclic group

Aliases: Dic105, C6.D35, C3⋊Dic35, C7⋊Dic15, C1055C4, C14.D15, C10.D21, C2.D105, C70.1S3, C42.1D5, C30.1D7, C211Dic5, C52Dic21, C153Dic7, C353Dic3, C210.1C2, SmallGroup(420,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C105 — Dic105
 Chief series C1 — C7 — C35 — C105 — C210 — Dic105
 Lower central C105 — Dic105
 Upper central C1 — C2

Generators and relations for Dic105
G = < a,b | a210=1, b2=a105, bab-1=a-1 >

Smallest permutation representation of Dic105
Regular action on 420 points
Generators in S420
(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)
(1 343 106 238)(2 342 107 237)(3 341 108 236)(4 340 109 235)(5 339 110 234)(6 338 111 233)(7 337 112 232)(8 336 113 231)(9 335 114 230)(10 334 115 229)(11 333 116 228)(12 332 117 227)(13 331 118 226)(14 330 119 225)(15 329 120 224)(16 328 121 223)(17 327 122 222)(18 326 123 221)(19 325 124 220)(20 324 125 219)(21 323 126 218)(22 322 127 217)(23 321 128 216)(24 320 129 215)(25 319 130 214)(26 318 131 213)(27 317 132 212)(28 316 133 211)(29 315 134 420)(30 314 135 419)(31 313 136 418)(32 312 137 417)(33 311 138 416)(34 310 139 415)(35 309 140 414)(36 308 141 413)(37 307 142 412)(38 306 143 411)(39 305 144 410)(40 304 145 409)(41 303 146 408)(42 302 147 407)(43 301 148 406)(44 300 149 405)(45 299 150 404)(46 298 151 403)(47 297 152 402)(48 296 153 401)(49 295 154 400)(50 294 155 399)(51 293 156 398)(52 292 157 397)(53 291 158 396)(54 290 159 395)(55 289 160 394)(56 288 161 393)(57 287 162 392)(58 286 163 391)(59 285 164 390)(60 284 165 389)(61 283 166 388)(62 282 167 387)(63 281 168 386)(64 280 169 385)(65 279 170 384)(66 278 171 383)(67 277 172 382)(68 276 173 381)(69 275 174 380)(70 274 175 379)(71 273 176 378)(72 272 177 377)(73 271 178 376)(74 270 179 375)(75 269 180 374)(76 268 181 373)(77 267 182 372)(78 266 183 371)(79 265 184 370)(80 264 185 369)(81 263 186 368)(82 262 187 367)(83 261 188 366)(84 260 189 365)(85 259 190 364)(86 258 191 363)(87 257 192 362)(88 256 193 361)(89 255 194 360)(90 254 195 359)(91 253 196 358)(92 252 197 357)(93 251 198 356)(94 250 199 355)(95 249 200 354)(96 248 201 353)(97 247 202 352)(98 246 203 351)(99 245 204 350)(100 244 205 349)(101 243 206 348)(102 242 207 347)(103 241 208 346)(104 240 209 345)(105 239 210 344)

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

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

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

108 conjugacy classes

 class 1 2 3 4A 4B 5A 5B 6 7A 7B 7C 10A 10B 14A 14B 14C 15A 15B 15C 15D 21A ··· 21F 30A 30B 30C 30D 35A ··· 35L 42A ··· 42F 70A ··· 70L 105A ··· 105X 210A ··· 210X order 1 2 3 4 4 5 5 6 7 7 7 10 10 14 14 14 15 15 15 15 21 ··· 21 30 30 30 30 35 ··· 35 42 ··· 42 70 ··· 70 105 ··· 105 210 ··· 210 size 1 1 2 105 105 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - - + + - + - - + - image C1 C2 C4 S3 D5 Dic3 D7 Dic5 Dic7 D15 D21 Dic15 D35 Dic21 Dic35 D105 Dic105 kernel Dic105 C210 C105 C70 C42 C35 C30 C21 C15 C14 C10 C7 C6 C5 C3 C2 C1 # reps 1 1 2 1 2 1 3 2 3 4 6 4 12 6 12 24 24

Matrix representation of Dic105 in GL4(𝔽421) generated by

 158 371 0 0 50 216 0 0 0 0 225 312 0 0 218 205
,
 79 167 0 0 8 342 0 0 0 0 215 99 0 0 222 206
G:=sub<GL(4,GF(421))| [158,50,0,0,371,216,0,0,0,0,225,218,0,0,312,205],[79,8,0,0,167,342,0,0,0,0,215,222,0,0,99,206] >;

Dic105 in GAP, Magma, Sage, TeX

{\rm Dic}_{105}
% in TeX

G:=Group("Dic105");
// GroupNames label

G:=SmallGroup(420,11);
// by ID

G=gap.SmallGroup(420,11);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-7,10,122,963,9004]);
// Polycyclic

G:=Group<a,b|a^210=1,b^2=a^105,b*a*b^-1=a^-1>;
// generators/relations

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