metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic67, C67⋊C4, C2.D67, C134.C2, SmallGroup(268,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C67 — Dic67 |
Generators and relations for Dic67
G = < a,b | a134=1, b2=a67, bab-1=a-1 >
Smallest permutation representation of Dic67
►Regular action on 268 points
Generators in S268
(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)
(1 152 68 219)(2 151 69 218)(3 150 70 217)(4 149 71 216)(5 148 72 215)(6 147 73 214)(7 146 74 213)(8 145 75 212)(9 144 76 211)(10 143 77 210)(11 142 78 209)(12 141 79 208)(13 140 80 207)(14 139 81 206)(15 138 82 205)(16 137 83 204)(17 136 84 203)(18 135 85 202)(19 268 86 201)(20 267 87 200)(21 266 88 199)(22 265 89 198)(23 264 90 197)(24 263 91 196)(25 262 92 195)(26 261 93 194)(27 260 94 193)(28 259 95 192)(29 258 96 191)(30 257 97 190)(31 256 98 189)(32 255 99 188)(33 254 100 187)(34 253 101 186)(35 252 102 185)(36 251 103 184)(37 250 104 183)(38 249 105 182)(39 248 106 181)(40 247 107 180)(41 246 108 179)(42 245 109 178)(43 244 110 177)(44 243 111 176)(45 242 112 175)(46 241 113 174)(47 240 114 173)(48 239 115 172)(49 238 116 171)(50 237 117 170)(51 236 118 169)(52 235 119 168)(53 234 120 167)(54 233 121 166)(55 232 122 165)(56 231 123 164)(57 230 124 163)(58 229 125 162)(59 228 126 161)(60 227 127 160)(61 226 128 159)(62 225 129 158)(63 224 130 157)(64 223 131 156)(65 222 132 155)(66 221 133 154)(67 220 134 153)
G:=sub<Sym(268)| (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), (1,152,68,219)(2,151,69,218)(3,150,70,217)(4,149,71,216)(5,148,72,215)(6,147,73,214)(7,146,74,213)(8,145,75,212)(9,144,76,211)(10,143,77,210)(11,142,78,209)(12,141,79,208)(13,140,80,207)(14,139,81,206)(15,138,82,205)(16,137,83,204)(17,136,84,203)(18,135,85,202)(19,268,86,201)(20,267,87,200)(21,266,88,199)(22,265,89,198)(23,264,90,197)(24,263,91,196)(25,262,92,195)(26,261,93,194)(27,260,94,193)(28,259,95,192)(29,258,96,191)(30,257,97,190)(31,256,98,189)(32,255,99,188)(33,254,100,187)(34,253,101,186)(35,252,102,185)(36,251,103,184)(37,250,104,183)(38,249,105,182)(39,248,106,181)(40,247,107,180)(41,246,108,179)(42,245,109,178)(43,244,110,177)(44,243,111,176)(45,242,112,175)(46,241,113,174)(47,240,114,173)(48,239,115,172)(49,238,116,171)(50,237,117,170)(51,236,118,169)(52,235,119,168)(53,234,120,167)(54,233,121,166)(55,232,122,165)(56,231,123,164)(57,230,124,163)(58,229,125,162)(59,228,126,161)(60,227,127,160)(61,226,128,159)(62,225,129,158)(63,224,130,157)(64,223,131,156)(65,222,132,155)(66,221,133,154)(67,220,134,153)>;
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), (1,152,68,219)(2,151,69,218)(3,150,70,217)(4,149,71,216)(5,148,72,215)(6,147,73,214)(7,146,74,213)(8,145,75,212)(9,144,76,211)(10,143,77,210)(11,142,78,209)(12,141,79,208)(13,140,80,207)(14,139,81,206)(15,138,82,205)(16,137,83,204)(17,136,84,203)(18,135,85,202)(19,268,86,201)(20,267,87,200)(21,266,88,199)(22,265,89,198)(23,264,90,197)(24,263,91,196)(25,262,92,195)(26,261,93,194)(27,260,94,193)(28,259,95,192)(29,258,96,191)(30,257,97,190)(31,256,98,189)(32,255,99,188)(33,254,100,187)(34,253,101,186)(35,252,102,185)(36,251,103,184)(37,250,104,183)(38,249,105,182)(39,248,106,181)(40,247,107,180)(41,246,108,179)(42,245,109,178)(43,244,110,177)(44,243,111,176)(45,242,112,175)(46,241,113,174)(47,240,114,173)(48,239,115,172)(49,238,116,171)(50,237,117,170)(51,236,118,169)(52,235,119,168)(53,234,120,167)(54,233,121,166)(55,232,122,165)(56,231,123,164)(57,230,124,163)(58,229,125,162)(59,228,126,161)(60,227,127,160)(61,226,128,159)(62,225,129,158)(63,224,130,157)(64,223,131,156)(65,222,132,155)(66,221,133,154)(67,220,134,153) );
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)], [(1,152,68,219),(2,151,69,218),(3,150,70,217),(4,149,71,216),(5,148,72,215),(6,147,73,214),(7,146,74,213),(8,145,75,212),(9,144,76,211),(10,143,77,210),(11,142,78,209),(12,141,79,208),(13,140,80,207),(14,139,81,206),(15,138,82,205),(16,137,83,204),(17,136,84,203),(18,135,85,202),(19,268,86,201),(20,267,87,200),(21,266,88,199),(22,265,89,198),(23,264,90,197),(24,263,91,196),(25,262,92,195),(26,261,93,194),(27,260,94,193),(28,259,95,192),(29,258,96,191),(30,257,97,190),(31,256,98,189),(32,255,99,188),(33,254,100,187),(34,253,101,186),(35,252,102,185),(36,251,103,184),(37,250,104,183),(38,249,105,182),(39,248,106,181),(40,247,107,180),(41,246,108,179),(42,245,109,178),(43,244,110,177),(44,243,111,176),(45,242,112,175),(46,241,113,174),(47,240,114,173),(48,239,115,172),(49,238,116,171),(50,237,117,170),(51,236,118,169),(52,235,119,168),(53,234,120,167),(54,233,121,166),(55,232,122,165),(56,231,123,164),(57,230,124,163),(58,229,125,162),(59,228,126,161),(60,227,127,160),(61,226,128,159),(62,225,129,158),(63,224,130,157),(64,223,131,156),(65,222,132,155),(66,221,133,154),(67,220,134,153)]])
70 conjugacy classes
| class | 1 | 2 | 4A | 4B | 67A | ··· | 67AG | 134A | ··· | 134AG |
| order | 1 | 2 | 4 | 4 | 67 | ··· | 67 | 134 | ··· | 134 |
| size | 1 | 1 | 67 | 67 | 2 | ··· | 2 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D67 | Dic67 |
| kernel | Dic67 | C134 | C67 | C2 | C1 |
| # reps | 1 | 1 | 2 | 33 | 33 |
Matrix representation of Dic67 ►in GL3(𝔽269) generated by
| 268 | 0 | 0 |
| 0 | 117 | 268 |
| 0 | 1 | 0 |
| 82 | 0 | 0 |
| 0 | 120 | 138 |
| 0 | 190 | 149 |
G:=sub<GL(3,GF(269))| [268,0,0,0,117,1,0,268,0],[82,0,0,0,120,190,0,138,149] >;
Dic67 in GAP, Magma, Sage, TeX
{\rm Dic}_{67} % in TeX
G:=Group("Dic67"); // GroupNames label
G:=SmallGroup(268,1);
// by ID
G=gap.SmallGroup(268,1);
# by ID
G:=PCGroup([3,-2,-2,-67,6,2378]);
// Polycyclic
G:=Group<a,b|a^134=1,b^2=a^67,b*a*b^-1=a^-1>;
// generators/relations
Export