Copied to
clipboard

G = Dic73order 292 = 22·73

Dicyclic group

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

Aliases: Dic73, C732C4, C2.D73, C146.C2, SmallGroup(292,1)

Series: Derived Chief Lower central Upper central

C1C73 — Dic73
C1C73C146 — Dic73
C73 — Dic73
C1C2

Generators and relations for Dic73
 G = < a,b | a146=1, b2=a73, bab-1=a-1 >

73C4

Smallest permutation representation of Dic73
Regular action on 292 points
Generators in S292
(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)
(1 241 74 168)(2 240 75 167)(3 239 76 166)(4 238 77 165)(5 237 78 164)(6 236 79 163)(7 235 80 162)(8 234 81 161)(9 233 82 160)(10 232 83 159)(11 231 84 158)(12 230 85 157)(13 229 86 156)(14 228 87 155)(15 227 88 154)(16 226 89 153)(17 225 90 152)(18 224 91 151)(19 223 92 150)(20 222 93 149)(21 221 94 148)(22 220 95 147)(23 219 96 292)(24 218 97 291)(25 217 98 290)(26 216 99 289)(27 215 100 288)(28 214 101 287)(29 213 102 286)(30 212 103 285)(31 211 104 284)(32 210 105 283)(33 209 106 282)(34 208 107 281)(35 207 108 280)(36 206 109 279)(37 205 110 278)(38 204 111 277)(39 203 112 276)(40 202 113 275)(41 201 114 274)(42 200 115 273)(43 199 116 272)(44 198 117 271)(45 197 118 270)(46 196 119 269)(47 195 120 268)(48 194 121 267)(49 193 122 266)(50 192 123 265)(51 191 124 264)(52 190 125 263)(53 189 126 262)(54 188 127 261)(55 187 128 260)(56 186 129 259)(57 185 130 258)(58 184 131 257)(59 183 132 256)(60 182 133 255)(61 181 134 254)(62 180 135 253)(63 179 136 252)(64 178 137 251)(65 177 138 250)(66 176 139 249)(67 175 140 248)(68 174 141 247)(69 173 142 246)(70 172 143 245)(71 171 144 244)(72 170 145 243)(73 169 146 242)

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

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

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

76 conjugacy classes

class 1  2 4A4B73A···73AJ146A···146AJ
order124473···73146···146
size1173732···22···2

76 irreducible representations

dim11122
type+++-
imageC1C2C4D73Dic73
kernelDic73C146C73C2C1
# reps1123636

Matrix representation of Dic73 in GL3(𝔽293) generated by

29200
0255292
010
,
13800
0127291
0153166
G:=sub<GL(3,GF(293))| [292,0,0,0,255,1,0,292,0],[138,0,0,0,127,153,0,291,166] >;

Dic73 in GAP, Magma, Sage, TeX

{\rm Dic}_{73}
% in TeX

G:=Group("Dic73");
// GroupNames label

G:=SmallGroup(292,1);
// by ID

G=gap.SmallGroup(292,1);
# by ID

G:=PCGroup([3,-2,-2,-73,6,2594]);
// Polycyclic

G:=Group<a,b|a^146=1,b^2=a^73,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic73 in TeX

׿
×
𝔽