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G = Dic90order 360 = 23·32·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

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

C1C90 — Dic90
C1C3C15C45C90Dic45 — Dic90
C45C90 — Dic90
C1C2C4

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

45C4
45C4
45Q8
15Dic3
15Dic3
9Dic5
9Dic5
15Dic6
5Dic9
5Dic9
9Dic10
3Dic15
3Dic15
5Dic18
3Dic30

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

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

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

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

93 conjugacy classes

class 1  2  3 4A4B4C5A5B 6 9A9B9C10A10B12A12B15A15B15C15D18A18B18C20A20B20C20D30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order1234445569991010121215151515181818202020203030303036···3645···4560···6090···90180···180
size1122909022222222222222222222222222···22···22···22···22···2

93 irreducible representations

dim1112222222222222222
type++++-++++-++-+-+-+-
imageC1C2C2S3Q8D5D6D9D10Dic6D15D18Dic10D30Dic18D45Dic30D90Dic90
kernelDic90Dic45C180C60C45C36C30C20C18C15C12C10C9C6C5C4C3C2C1
# reps1211121322434461281224

Matrix representation of Dic90 in GL2(𝔽181) generated by

65158
2342
,
149107
7532
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

Subgroup lattice of Dic90 in TeX

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