Copied to
clipboard

## G = Dic90order 360 = 23·32·5

### Dicyclic group

Aliases: Dic90, C4.D45, C452Q8, C60.1S3, C20.1D9, C6.8D30, C36.1D5, C2.3D90, C52Dic18, C92Dic10, C3.Dic30, C180.1C2, C12.1D15, C18.8D10, C10.8D18, C30.40D6, C90.8C22, C15.2Dic6, Dic45.1C2, SmallGroup(360,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C90 — Dic90
 Chief series C1 — C3 — C15 — C45 — C90 — Dic45 — Dic90
 Lower central C45 — C90 — Dic90
 Upper central C1 — C2 — C4

Generators and relations for Dic90
G = < a,b | a180=1, b2=a90, bab-1=a-1 >

Smallest permutation representation of Dic90
Regular action on 360 points
Generators in S360
(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)
(1 238 91 328)(2 237 92 327)(3 236 93 326)(4 235 94 325)(5 234 95 324)(6 233 96 323)(7 232 97 322)(8 231 98 321)(9 230 99 320)(10 229 100 319)(11 228 101 318)(12 227 102 317)(13 226 103 316)(14 225 104 315)(15 224 105 314)(16 223 106 313)(17 222 107 312)(18 221 108 311)(19 220 109 310)(20 219 110 309)(21 218 111 308)(22 217 112 307)(23 216 113 306)(24 215 114 305)(25 214 115 304)(26 213 116 303)(27 212 117 302)(28 211 118 301)(29 210 119 300)(30 209 120 299)(31 208 121 298)(32 207 122 297)(33 206 123 296)(34 205 124 295)(35 204 125 294)(36 203 126 293)(37 202 127 292)(38 201 128 291)(39 200 129 290)(40 199 130 289)(41 198 131 288)(42 197 132 287)(43 196 133 286)(44 195 134 285)(45 194 135 284)(46 193 136 283)(47 192 137 282)(48 191 138 281)(49 190 139 280)(50 189 140 279)(51 188 141 278)(52 187 142 277)(53 186 143 276)(54 185 144 275)(55 184 145 274)(56 183 146 273)(57 182 147 272)(58 181 148 271)(59 360 149 270)(60 359 150 269)(61 358 151 268)(62 357 152 267)(63 356 153 266)(64 355 154 265)(65 354 155 264)(66 353 156 263)(67 352 157 262)(68 351 158 261)(69 350 159 260)(70 349 160 259)(71 348 161 258)(72 347 162 257)(73 346 163 256)(74 345 164 255)(75 344 165 254)(76 343 166 253)(77 342 167 252)(78 341 168 251)(79 340 169 250)(80 339 170 249)(81 338 171 248)(82 337 172 247)(83 336 173 246)(84 335 174 245)(85 334 175 244)(86 333 176 243)(87 332 177 242)(88 331 178 241)(89 330 179 240)(90 329 180 239)

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

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

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

93 conjugacy classes

 class 1 2 3 4A 4B 4C 5A 5B 6 9A 9B 9C 10A 10B 12A 12B 15A 15B 15C 15D 18A 18B 18C 20A 20B 20C 20D 30A 30B 30C 30D 36A ··· 36F 45A ··· 45L 60A ··· 60H 90A ··· 90L 180A ··· 180X order 1 2 3 4 4 4 5 5 6 9 9 9 10 10 12 12 15 15 15 15 18 18 18 20 20 20 20 30 30 30 30 36 ··· 36 45 ··· 45 60 ··· 60 90 ··· 90 180 ··· 180 size 1 1 2 2 90 90 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 2 ··· 2 2 ··· 2

93 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + + + - + + - + - + - + - image C1 C2 C2 S3 Q8 D5 D6 D9 D10 Dic6 D15 D18 Dic10 D30 Dic18 D45 Dic30 D90 Dic90 kernel Dic90 Dic45 C180 C60 C45 C36 C30 C20 C18 C15 C12 C10 C9 C6 C5 C4 C3 C2 C1 # reps 1 2 1 1 1 2 1 3 2 2 4 3 4 4 6 12 8 12 24

Matrix representation of Dic90 in GL2(𝔽181) generated by

 65 158 23 42
,
 149 107 75 32
G:=sub<GL(2,GF(181))| [65,23,158,42],[149,75,107,32] >;

Dic90 in GAP, Magma, Sage, TeX

{\rm Dic}_{90}
% in TeX

G:=Group("Dic90");
// GroupNames label

G:=SmallGroup(360,25);
// by ID

G=gap.SmallGroup(360,25);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,73,31,3267,741,2884,8645]);
// Polycyclic

G:=Group<a,b|a^180=1,b^2=a^90,b*a*b^-1=a^-1>;
// generators/relations

Export

׿
×
𝔽