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G = Dic82order 328 = 23·41

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic82, C41⋊Q8, C4.D41, C2.3D82, C164.1C2, C82.1C22, Dic41.1C2, SmallGroup(328,4)

Series: Derived Chief Lower central Upper central

C1C82 — Dic82
C1C41C82Dic41 — Dic82
C41C82 — Dic82
C1C2C4

Generators and relations for Dic82
 G = < a,b | a164=1, b2=a82, bab-1=a-1 >

41C4
41C4
41Q8

Smallest permutation representation of Dic82
Regular action on 328 points
Generators in S328
(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)
(1 224 83 306)(2 223 84 305)(3 222 85 304)(4 221 86 303)(5 220 87 302)(6 219 88 301)(7 218 89 300)(8 217 90 299)(9 216 91 298)(10 215 92 297)(11 214 93 296)(12 213 94 295)(13 212 95 294)(14 211 96 293)(15 210 97 292)(16 209 98 291)(17 208 99 290)(18 207 100 289)(19 206 101 288)(20 205 102 287)(21 204 103 286)(22 203 104 285)(23 202 105 284)(24 201 106 283)(25 200 107 282)(26 199 108 281)(27 198 109 280)(28 197 110 279)(29 196 111 278)(30 195 112 277)(31 194 113 276)(32 193 114 275)(33 192 115 274)(34 191 116 273)(35 190 117 272)(36 189 118 271)(37 188 119 270)(38 187 120 269)(39 186 121 268)(40 185 122 267)(41 184 123 266)(42 183 124 265)(43 182 125 264)(44 181 126 263)(45 180 127 262)(46 179 128 261)(47 178 129 260)(48 177 130 259)(49 176 131 258)(50 175 132 257)(51 174 133 256)(52 173 134 255)(53 172 135 254)(54 171 136 253)(55 170 137 252)(56 169 138 251)(57 168 139 250)(58 167 140 249)(59 166 141 248)(60 165 142 247)(61 328 143 246)(62 327 144 245)(63 326 145 244)(64 325 146 243)(65 324 147 242)(66 323 148 241)(67 322 149 240)(68 321 150 239)(69 320 151 238)(70 319 152 237)(71 318 153 236)(72 317 154 235)(73 316 155 234)(74 315 156 233)(75 314 157 232)(76 313 158 231)(77 312 159 230)(78 311 160 229)(79 310 161 228)(80 309 162 227)(81 308 163 226)(82 307 164 225)

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

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

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

85 conjugacy classes

class 1  2 4A4B4C41A···41T82A···82T164A···164AN
order1244441···4182···82164···164
size11282822···22···22···2

85 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D41D82Dic82
kernelDic82Dic41C164C41C4C2C1
# reps1211202040

Matrix representation of Dic82 in GL2(𝔽821) generated by

5323
76165
,
362553
780459
G:=sub<GL(2,GF(821))| [5,761,323,65],[362,780,553,459] >;

Dic82 in GAP, Magma, Sage, TeX

{\rm Dic}_{82}
% in TeX

G:=Group("Dic82");
// GroupNames label

G:=SmallGroup(328,4);
// by ID

G=gap.SmallGroup(328,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,16,49,21,5123]);
// Polycyclic

G:=Group<a,b|a^164=1,b^2=a^82,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic82 in TeX

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