metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic69, C69⋊1C4, C46.S3, C6.D23, C2.D69, C3⋊Dic23, C23⋊Dic3, C138.1C2, SmallGroup(276,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C69 — Dic69 |
Generators and relations for Dic69
G = < a,b | a138=1, b2=a69, bab-1=a-1 >
Smallest permutation representation of Dic69
►Regular action on 276 points
Generators in S276
(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)
(1 170 70 239)(2 169 71 238)(3 168 72 237)(4 167 73 236)(5 166 74 235)(6 165 75 234)(7 164 76 233)(8 163 77 232)(9 162 78 231)(10 161 79 230)(11 160 80 229)(12 159 81 228)(13 158 82 227)(14 157 83 226)(15 156 84 225)(16 155 85 224)(17 154 86 223)(18 153 87 222)(19 152 88 221)(20 151 89 220)(21 150 90 219)(22 149 91 218)(23 148 92 217)(24 147 93 216)(25 146 94 215)(26 145 95 214)(27 144 96 213)(28 143 97 212)(29 142 98 211)(30 141 99 210)(31 140 100 209)(32 139 101 208)(33 276 102 207)(34 275 103 206)(35 274 104 205)(36 273 105 204)(37 272 106 203)(38 271 107 202)(39 270 108 201)(40 269 109 200)(41 268 110 199)(42 267 111 198)(43 266 112 197)(44 265 113 196)(45 264 114 195)(46 263 115 194)(47 262 116 193)(48 261 117 192)(49 260 118 191)(50 259 119 190)(51 258 120 189)(52 257 121 188)(53 256 122 187)(54 255 123 186)(55 254 124 185)(56 253 125 184)(57 252 126 183)(58 251 127 182)(59 250 128 181)(60 249 129 180)(61 248 130 179)(62 247 131 178)(63 246 132 177)(64 245 133 176)(65 244 134 175)(66 243 135 174)(67 242 136 173)(68 241 137 172)(69 240 138 171)
G:=sub<Sym(276)| (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), (1,170,70,239)(2,169,71,238)(3,168,72,237)(4,167,73,236)(5,166,74,235)(6,165,75,234)(7,164,76,233)(8,163,77,232)(9,162,78,231)(10,161,79,230)(11,160,80,229)(12,159,81,228)(13,158,82,227)(14,157,83,226)(15,156,84,225)(16,155,85,224)(17,154,86,223)(18,153,87,222)(19,152,88,221)(20,151,89,220)(21,150,90,219)(22,149,91,218)(23,148,92,217)(24,147,93,216)(25,146,94,215)(26,145,95,214)(27,144,96,213)(28,143,97,212)(29,142,98,211)(30,141,99,210)(31,140,100,209)(32,139,101,208)(33,276,102,207)(34,275,103,206)(35,274,104,205)(36,273,105,204)(37,272,106,203)(38,271,107,202)(39,270,108,201)(40,269,109,200)(41,268,110,199)(42,267,111,198)(43,266,112,197)(44,265,113,196)(45,264,114,195)(46,263,115,194)(47,262,116,193)(48,261,117,192)(49,260,118,191)(50,259,119,190)(51,258,120,189)(52,257,121,188)(53,256,122,187)(54,255,123,186)(55,254,124,185)(56,253,125,184)(57,252,126,183)(58,251,127,182)(59,250,128,181)(60,249,129,180)(61,248,130,179)(62,247,131,178)(63,246,132,177)(64,245,133,176)(65,244,134,175)(66,243,135,174)(67,242,136,173)(68,241,137,172)(69,240,138,171)>;
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), (1,170,70,239)(2,169,71,238)(3,168,72,237)(4,167,73,236)(5,166,74,235)(6,165,75,234)(7,164,76,233)(8,163,77,232)(9,162,78,231)(10,161,79,230)(11,160,80,229)(12,159,81,228)(13,158,82,227)(14,157,83,226)(15,156,84,225)(16,155,85,224)(17,154,86,223)(18,153,87,222)(19,152,88,221)(20,151,89,220)(21,150,90,219)(22,149,91,218)(23,148,92,217)(24,147,93,216)(25,146,94,215)(26,145,95,214)(27,144,96,213)(28,143,97,212)(29,142,98,211)(30,141,99,210)(31,140,100,209)(32,139,101,208)(33,276,102,207)(34,275,103,206)(35,274,104,205)(36,273,105,204)(37,272,106,203)(38,271,107,202)(39,270,108,201)(40,269,109,200)(41,268,110,199)(42,267,111,198)(43,266,112,197)(44,265,113,196)(45,264,114,195)(46,263,115,194)(47,262,116,193)(48,261,117,192)(49,260,118,191)(50,259,119,190)(51,258,120,189)(52,257,121,188)(53,256,122,187)(54,255,123,186)(55,254,124,185)(56,253,125,184)(57,252,126,183)(58,251,127,182)(59,250,128,181)(60,249,129,180)(61,248,130,179)(62,247,131,178)(63,246,132,177)(64,245,133,176)(65,244,134,175)(66,243,135,174)(67,242,136,173)(68,241,137,172)(69,240,138,171) );
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)], [(1,170,70,239),(2,169,71,238),(3,168,72,237),(4,167,73,236),(5,166,74,235),(6,165,75,234),(7,164,76,233),(8,163,77,232),(9,162,78,231),(10,161,79,230),(11,160,80,229),(12,159,81,228),(13,158,82,227),(14,157,83,226),(15,156,84,225),(16,155,85,224),(17,154,86,223),(18,153,87,222),(19,152,88,221),(20,151,89,220),(21,150,90,219),(22,149,91,218),(23,148,92,217),(24,147,93,216),(25,146,94,215),(26,145,95,214),(27,144,96,213),(28,143,97,212),(29,142,98,211),(30,141,99,210),(31,140,100,209),(32,139,101,208),(33,276,102,207),(34,275,103,206),(35,274,104,205),(36,273,105,204),(37,272,106,203),(38,271,107,202),(39,270,108,201),(40,269,109,200),(41,268,110,199),(42,267,111,198),(43,266,112,197),(44,265,113,196),(45,264,114,195),(46,263,115,194),(47,262,116,193),(48,261,117,192),(49,260,118,191),(50,259,119,190),(51,258,120,189),(52,257,121,188),(53,256,122,187),(54,255,123,186),(55,254,124,185),(56,253,125,184),(57,252,126,183),(58,251,127,182),(59,250,128,181),(60,249,129,180),(61,248,130,179),(62,247,131,178),(63,246,132,177),(64,245,133,176),(65,244,134,175),(66,243,135,174),(67,242,136,173),(68,241,137,172),(69,240,138,171)]])
72 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 23A | ··· | 23K | 46A | ··· | 46K | 69A | ··· | 69V | 138A | ··· | 138V |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 2 | 69 | 69 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D23 | Dic23 | D69 | Dic69 |
| kernel | Dic69 | C138 | C69 | C46 | C23 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 11 | 11 | 22 | 22 |
Matrix representation of Dic69 ►in GL3(𝔽277) generated by
| 276 | 0 | 0 |
| 0 | 17 | 15 |
| 0 | 262 | 52 |
| 60 | 0 | 0 |
| 0 | 222 | 83 |
| 0 | 47 | 55 |
G:=sub<GL(3,GF(277))| [276,0,0,0,17,262,0,15,52],[60,0,0,0,222,47,0,83,55] >;
Dic69 in GAP, Magma, Sage, TeX
{\rm Dic}_{69} % in TeX
G:=Group("Dic69"); // GroupNames label
G:=SmallGroup(276,3);
// by ID
G=gap.SmallGroup(276,3);
# by ID
G:=PCGroup([4,-2,-2,-3,-23,8,98,4227]);
// Polycyclic
G:=Group<a,b|a^138=1,b^2=a^69,b*a*b^-1=a^-1>;
// generators/relations
Export