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G = Dic86order 344 = 23·43

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic86, C43⋊Q8, C4.D43, C2.3D86, C172.1C2, Dic43.C2, C86.1C22, SmallGroup(344,3)

Series: Derived Chief Lower central Upper central

C1C86 — Dic86
C1C43C86Dic43 — Dic86
C43C86 — Dic86
C1C2C4

Generators and relations for Dic86
 G = < a,b | a172=1, b2=a86, bab-1=a-1 >

43C4
43C4
43Q8

Smallest permutation representation of Dic86
Regular action on 344 points
Generators in S344
(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)
(1 245 87 331)(2 244 88 330)(3 243 89 329)(4 242 90 328)(5 241 91 327)(6 240 92 326)(7 239 93 325)(8 238 94 324)(9 237 95 323)(10 236 96 322)(11 235 97 321)(12 234 98 320)(13 233 99 319)(14 232 100 318)(15 231 101 317)(16 230 102 316)(17 229 103 315)(18 228 104 314)(19 227 105 313)(20 226 106 312)(21 225 107 311)(22 224 108 310)(23 223 109 309)(24 222 110 308)(25 221 111 307)(26 220 112 306)(27 219 113 305)(28 218 114 304)(29 217 115 303)(30 216 116 302)(31 215 117 301)(32 214 118 300)(33 213 119 299)(34 212 120 298)(35 211 121 297)(36 210 122 296)(37 209 123 295)(38 208 124 294)(39 207 125 293)(40 206 126 292)(41 205 127 291)(42 204 128 290)(43 203 129 289)(44 202 130 288)(45 201 131 287)(46 200 132 286)(47 199 133 285)(48 198 134 284)(49 197 135 283)(50 196 136 282)(51 195 137 281)(52 194 138 280)(53 193 139 279)(54 192 140 278)(55 191 141 277)(56 190 142 276)(57 189 143 275)(58 188 144 274)(59 187 145 273)(60 186 146 272)(61 185 147 271)(62 184 148 270)(63 183 149 269)(64 182 150 268)(65 181 151 267)(66 180 152 266)(67 179 153 265)(68 178 154 264)(69 177 155 263)(70 176 156 262)(71 175 157 261)(72 174 158 260)(73 173 159 259)(74 344 160 258)(75 343 161 257)(76 342 162 256)(77 341 163 255)(78 340 164 254)(79 339 165 253)(80 338 166 252)(81 337 167 251)(82 336 168 250)(83 335 169 249)(84 334 170 248)(85 333 171 247)(86 332 172 246)

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

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

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

89 conjugacy classes

class 1  2 4A4B4C43A···43U86A···86U172A···172AP
order1244443···4386···86172···172
size11286862···22···22···2

89 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D43D86Dic86
kernelDic86Dic43C172C43C4C2C1
# reps1211212142

Matrix representation of Dic86 in GL2(𝔽173) generated by

19116
5784
,
3253
59141
G:=sub<GL(2,GF(173))| [19,57,116,84],[32,59,53,141] >;

Dic86 in GAP, Magma, Sage, TeX

{\rm Dic}_{86}
% in TeX

G:=Group("Dic86");
// GroupNames label

G:=SmallGroup(344,3);
// by ID

G=gap.SmallGroup(344,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-43,16,49,21,5379]);
// Polycyclic

G:=Group<a,b|a^172=1,b^2=a^86,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic86 in TeX

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