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G = Dic83order 332 = 22·83

Dicyclic group

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

Aliases: Dic83, C83⋊C4, C2.D83, C166.C2, SmallGroup(332,1)

Series: Derived Chief Lower central Upper central

C1C83 — Dic83
C1C83C166 — Dic83
C83 — Dic83
C1C2

Generators and relations for Dic83
 G = < a,b | a166=1, b2=a83, bab-1=a-1 >

83C4

Smallest permutation representation of Dic83
Regular action on 332 points
Generators in S332
(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)
(1 201 84 284)(2 200 85 283)(3 199 86 282)(4 198 87 281)(5 197 88 280)(6 196 89 279)(7 195 90 278)(8 194 91 277)(9 193 92 276)(10 192 93 275)(11 191 94 274)(12 190 95 273)(13 189 96 272)(14 188 97 271)(15 187 98 270)(16 186 99 269)(17 185 100 268)(18 184 101 267)(19 183 102 266)(20 182 103 265)(21 181 104 264)(22 180 105 263)(23 179 106 262)(24 178 107 261)(25 177 108 260)(26 176 109 259)(27 175 110 258)(28 174 111 257)(29 173 112 256)(30 172 113 255)(31 171 114 254)(32 170 115 253)(33 169 116 252)(34 168 117 251)(35 167 118 250)(36 332 119 249)(37 331 120 248)(38 330 121 247)(39 329 122 246)(40 328 123 245)(41 327 124 244)(42 326 125 243)(43 325 126 242)(44 324 127 241)(45 323 128 240)(46 322 129 239)(47 321 130 238)(48 320 131 237)(49 319 132 236)(50 318 133 235)(51 317 134 234)(52 316 135 233)(53 315 136 232)(54 314 137 231)(55 313 138 230)(56 312 139 229)(57 311 140 228)(58 310 141 227)(59 309 142 226)(60 308 143 225)(61 307 144 224)(62 306 145 223)(63 305 146 222)(64 304 147 221)(65 303 148 220)(66 302 149 219)(67 301 150 218)(68 300 151 217)(69 299 152 216)(70 298 153 215)(71 297 154 214)(72 296 155 213)(73 295 156 212)(74 294 157 211)(75 293 158 210)(76 292 159 209)(77 291 160 208)(78 290 161 207)(79 289 162 206)(80 288 163 205)(81 287 164 204)(82 286 165 203)(83 285 166 202)

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

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

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

86 conjugacy classes

class 1  2 4A4B83A···83AO166A···166AO
order124483···83166···166
size1183832···22···2

86 irreducible representations

dim11122
type+++-
imageC1C2C4D83Dic83
kernelDic83C166C83C2C1
# reps1124141

Matrix representation of Dic83 in GL2(𝔽997) generated by

4201
9960
,
362666
170635
G:=sub<GL(2,GF(997))| [420,996,1,0],[362,170,666,635] >;

Dic83 in GAP, Magma, Sage, TeX

{\rm Dic}_{83}
% in TeX

G:=Group("Dic83");
// GroupNames label

G:=SmallGroup(332,1);
// by ID

G=gap.SmallGroup(332,1);
# by ID

G:=PCGroup([3,-2,-2,-83,6,2954]);
// Polycyclic

G:=Group<a,b|a^166=1,b^2=a^83,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic83 in TeX

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