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G = Dic89order 356 = 22·89

Dicyclic group

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

Aliases: Dic89, C892C4, C2.D89, C178.C2, SmallGroup(356,1)

Series: Derived Chief Lower central Upper central

C1C89 — Dic89
C1C89C178 — Dic89
C89 — Dic89
C1C2

Generators and relations for Dic89
 G = < a,b | a178=1, b2=a89, bab-1=a-1 >

89C4

Smallest permutation representation of Dic89
Regular action on 356 points
Generators in S356
(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)
(1 280 90 191)(2 279 91 190)(3 278 92 189)(4 277 93 188)(5 276 94 187)(6 275 95 186)(7 274 96 185)(8 273 97 184)(9 272 98 183)(10 271 99 182)(11 270 100 181)(12 269 101 180)(13 268 102 179)(14 267 103 356)(15 266 104 355)(16 265 105 354)(17 264 106 353)(18 263 107 352)(19 262 108 351)(20 261 109 350)(21 260 110 349)(22 259 111 348)(23 258 112 347)(24 257 113 346)(25 256 114 345)(26 255 115 344)(27 254 116 343)(28 253 117 342)(29 252 118 341)(30 251 119 340)(31 250 120 339)(32 249 121 338)(33 248 122 337)(34 247 123 336)(35 246 124 335)(36 245 125 334)(37 244 126 333)(38 243 127 332)(39 242 128 331)(40 241 129 330)(41 240 130 329)(42 239 131 328)(43 238 132 327)(44 237 133 326)(45 236 134 325)(46 235 135 324)(47 234 136 323)(48 233 137 322)(49 232 138 321)(50 231 139 320)(51 230 140 319)(52 229 141 318)(53 228 142 317)(54 227 143 316)(55 226 144 315)(56 225 145 314)(57 224 146 313)(58 223 147 312)(59 222 148 311)(60 221 149 310)(61 220 150 309)(62 219 151 308)(63 218 152 307)(64 217 153 306)(65 216 154 305)(66 215 155 304)(67 214 156 303)(68 213 157 302)(69 212 158 301)(70 211 159 300)(71 210 160 299)(72 209 161 298)(73 208 162 297)(74 207 163 296)(75 206 164 295)(76 205 165 294)(77 204 166 293)(78 203 167 292)(79 202 168 291)(80 201 169 290)(81 200 170 289)(82 199 171 288)(83 198 172 287)(84 197 173 286)(85 196 174 285)(86 195 175 284)(87 194 176 283)(88 193 177 282)(89 192 178 281)

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

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

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

92 conjugacy classes

class 1  2 4A4B89A···89AR178A···178AR
order124489···89178···178
size1189892···22···2

92 irreducible representations

dim11122
type+++-
imageC1C2C4D89Dic89
kernelDic89C178C89C2C1
# reps1124444

Matrix representation of Dic89 in GL2(𝔽1069) generated by

111
10680
,
487430
418582
G:=sub<GL(2,GF(1069))| [11,1068,1,0],[487,418,430,582] >;

Dic89 in GAP, Magma, Sage, TeX

{\rm Dic}_{89}
% in TeX

G:=Group("Dic89");
// GroupNames label

G:=SmallGroup(356,1);
// by ID

G=gap.SmallGroup(356,1);
# by ID

G:=PCGroup([3,-2,-2,-89,6,3170]);
// Polycyclic

G:=Group<a,b|a^178=1,b^2=a^89,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic89 in TeX

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