metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic73, C73⋊2C4, C2.D73, C146.C2, SmallGroup(292,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C73 — Dic73 |
Generators and relations for Dic73
G = < a,b | a146=1, b2=a73, bab-1=a-1 >
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)
(1 209 74 282)(2 208 75 281)(3 207 76 280)(4 206 77 279)(5 205 78 278)(6 204 79 277)(7 203 80 276)(8 202 81 275)(9 201 82 274)(10 200 83 273)(11 199 84 272)(12 198 85 271)(13 197 86 270)(14 196 87 269)(15 195 88 268)(16 194 89 267)(17 193 90 266)(18 192 91 265)(19 191 92 264)(20 190 93 263)(21 189 94 262)(22 188 95 261)(23 187 96 260)(24 186 97 259)(25 185 98 258)(26 184 99 257)(27 183 100 256)(28 182 101 255)(29 181 102 254)(30 180 103 253)(31 179 104 252)(32 178 105 251)(33 177 106 250)(34 176 107 249)(35 175 108 248)(36 174 109 247)(37 173 110 246)(38 172 111 245)(39 171 112 244)(40 170 113 243)(41 169 114 242)(42 168 115 241)(43 167 116 240)(44 166 117 239)(45 165 118 238)(46 164 119 237)(47 163 120 236)(48 162 121 235)(49 161 122 234)(50 160 123 233)(51 159 124 232)(52 158 125 231)(53 157 126 230)(54 156 127 229)(55 155 128 228)(56 154 129 227)(57 153 130 226)(58 152 131 225)(59 151 132 224)(60 150 133 223)(61 149 134 222)(62 148 135 221)(63 147 136 220)(64 292 137 219)(65 291 138 218)(66 290 139 217)(67 289 140 216)(68 288 141 215)(69 287 142 214)(70 286 143 213)(71 285 144 212)(72 284 145 211)(73 283 146 210)
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), (1,209,74,282)(2,208,75,281)(3,207,76,280)(4,206,77,279)(5,205,78,278)(6,204,79,277)(7,203,80,276)(8,202,81,275)(9,201,82,274)(10,200,83,273)(11,199,84,272)(12,198,85,271)(13,197,86,270)(14,196,87,269)(15,195,88,268)(16,194,89,267)(17,193,90,266)(18,192,91,265)(19,191,92,264)(20,190,93,263)(21,189,94,262)(22,188,95,261)(23,187,96,260)(24,186,97,259)(25,185,98,258)(26,184,99,257)(27,183,100,256)(28,182,101,255)(29,181,102,254)(30,180,103,253)(31,179,104,252)(32,178,105,251)(33,177,106,250)(34,176,107,249)(35,175,108,248)(36,174,109,247)(37,173,110,246)(38,172,111,245)(39,171,112,244)(40,170,113,243)(41,169,114,242)(42,168,115,241)(43,167,116,240)(44,166,117,239)(45,165,118,238)(46,164,119,237)(47,163,120,236)(48,162,121,235)(49,161,122,234)(50,160,123,233)(51,159,124,232)(52,158,125,231)(53,157,126,230)(54,156,127,229)(55,155,128,228)(56,154,129,227)(57,153,130,226)(58,152,131,225)(59,151,132,224)(60,150,133,223)(61,149,134,222)(62,148,135,221)(63,147,136,220)(64,292,137,219)(65,291,138,218)(66,290,139,217)(67,289,140,216)(68,288,141,215)(69,287,142,214)(70,286,143,213)(71,285,144,212)(72,284,145,211)(73,283,146,210)>;
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), (1,209,74,282)(2,208,75,281)(3,207,76,280)(4,206,77,279)(5,205,78,278)(6,204,79,277)(7,203,80,276)(8,202,81,275)(9,201,82,274)(10,200,83,273)(11,199,84,272)(12,198,85,271)(13,197,86,270)(14,196,87,269)(15,195,88,268)(16,194,89,267)(17,193,90,266)(18,192,91,265)(19,191,92,264)(20,190,93,263)(21,189,94,262)(22,188,95,261)(23,187,96,260)(24,186,97,259)(25,185,98,258)(26,184,99,257)(27,183,100,256)(28,182,101,255)(29,181,102,254)(30,180,103,253)(31,179,104,252)(32,178,105,251)(33,177,106,250)(34,176,107,249)(35,175,108,248)(36,174,109,247)(37,173,110,246)(38,172,111,245)(39,171,112,244)(40,170,113,243)(41,169,114,242)(42,168,115,241)(43,167,116,240)(44,166,117,239)(45,165,118,238)(46,164,119,237)(47,163,120,236)(48,162,121,235)(49,161,122,234)(50,160,123,233)(51,159,124,232)(52,158,125,231)(53,157,126,230)(54,156,127,229)(55,155,128,228)(56,154,129,227)(57,153,130,226)(58,152,131,225)(59,151,132,224)(60,150,133,223)(61,149,134,222)(62,148,135,221)(63,147,136,220)(64,292,137,219)(65,291,138,218)(66,290,139,217)(67,289,140,216)(68,288,141,215)(69,287,142,214)(70,286,143,213)(71,285,144,212)(72,284,145,211)(73,283,146,210) );
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)], [(1,209,74,282),(2,208,75,281),(3,207,76,280),(4,206,77,279),(5,205,78,278),(6,204,79,277),(7,203,80,276),(8,202,81,275),(9,201,82,274),(10,200,83,273),(11,199,84,272),(12,198,85,271),(13,197,86,270),(14,196,87,269),(15,195,88,268),(16,194,89,267),(17,193,90,266),(18,192,91,265),(19,191,92,264),(20,190,93,263),(21,189,94,262),(22,188,95,261),(23,187,96,260),(24,186,97,259),(25,185,98,258),(26,184,99,257),(27,183,100,256),(28,182,101,255),(29,181,102,254),(30,180,103,253),(31,179,104,252),(32,178,105,251),(33,177,106,250),(34,176,107,249),(35,175,108,248),(36,174,109,247),(37,173,110,246),(38,172,111,245),(39,171,112,244),(40,170,113,243),(41,169,114,242),(42,168,115,241),(43,167,116,240),(44,166,117,239),(45,165,118,238),(46,164,119,237),(47,163,120,236),(48,162,121,235),(49,161,122,234),(50,160,123,233),(51,159,124,232),(52,158,125,231),(53,157,126,230),(54,156,127,229),(55,155,128,228),(56,154,129,227),(57,153,130,226),(58,152,131,225),(59,151,132,224),(60,150,133,223),(61,149,134,222),(62,148,135,221),(63,147,136,220),(64,292,137,219),(65,291,138,218),(66,290,139,217),(67,289,140,216),(68,288,141,215),(69,287,142,214),(70,286,143,213),(71,285,144,212),(72,284,145,211),(73,283,146,210)]])
76 conjugacy classes
| class | 1 | 2 | 4A | 4B | 73A | ··· | 73AJ | 146A | ··· | 146AJ |
| order | 1 | 2 | 4 | 4 | 73 | ··· | 73 | 146 | ··· | 146 |
| size | 1 | 1 | 73 | 73 | 2 | ··· | 2 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D73 | Dic73 |
| kernel | Dic73 | C146 | C73 | C2 | C1 |
| # reps | 1 | 1 | 2 | 36 | 36 |
Matrix representation of Dic73 ►in GL3(𝔽293) generated by
| 292 | 0 | 0 |
| 0 | 255 | 292 |
| 0 | 1 | 0 |
| 138 | 0 | 0 |
| 0 | 127 | 291 |
| 0 | 153 | 166 |
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
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