metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic68, C8.D17, C17⋊1Q16, C2.5D68, C34.3D4, C136.1C2, C4.10D34, C68.10C22, Dic34.1C2, SmallGroup(272,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic68
G = < a,b | a136=1, b2=a68, bab-1=a-1 >
Smallest permutation representation of Dic68
►Regular action on 272 points
Generators in S272
(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)
(1 153 69 221)(2 152 70 220)(3 151 71 219)(4 150 72 218)(5 149 73 217)(6 148 74 216)(7 147 75 215)(8 146 76 214)(9 145 77 213)(10 144 78 212)(11 143 79 211)(12 142 80 210)(13 141 81 209)(14 140 82 208)(15 139 83 207)(16 138 84 206)(17 137 85 205)(18 272 86 204)(19 271 87 203)(20 270 88 202)(21 269 89 201)(22 268 90 200)(23 267 91 199)(24 266 92 198)(25 265 93 197)(26 264 94 196)(27 263 95 195)(28 262 96 194)(29 261 97 193)(30 260 98 192)(31 259 99 191)(32 258 100 190)(33 257 101 189)(34 256 102 188)(35 255 103 187)(36 254 104 186)(37 253 105 185)(38 252 106 184)(39 251 107 183)(40 250 108 182)(41 249 109 181)(42 248 110 180)(43 247 111 179)(44 246 112 178)(45 245 113 177)(46 244 114 176)(47 243 115 175)(48 242 116 174)(49 241 117 173)(50 240 118 172)(51 239 119 171)(52 238 120 170)(53 237 121 169)(54 236 122 168)(55 235 123 167)(56 234 124 166)(57 233 125 165)(58 232 126 164)(59 231 127 163)(60 230 128 162)(61 229 129 161)(62 228 130 160)(63 227 131 159)(64 226 132 158)(65 225 133 157)(66 224 134 156)(67 223 135 155)(68 222 136 154)
G:=sub<Sym(272)| (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), (1,153,69,221)(2,152,70,220)(3,151,71,219)(4,150,72,218)(5,149,73,217)(6,148,74,216)(7,147,75,215)(8,146,76,214)(9,145,77,213)(10,144,78,212)(11,143,79,211)(12,142,80,210)(13,141,81,209)(14,140,82,208)(15,139,83,207)(16,138,84,206)(17,137,85,205)(18,272,86,204)(19,271,87,203)(20,270,88,202)(21,269,89,201)(22,268,90,200)(23,267,91,199)(24,266,92,198)(25,265,93,197)(26,264,94,196)(27,263,95,195)(28,262,96,194)(29,261,97,193)(30,260,98,192)(31,259,99,191)(32,258,100,190)(33,257,101,189)(34,256,102,188)(35,255,103,187)(36,254,104,186)(37,253,105,185)(38,252,106,184)(39,251,107,183)(40,250,108,182)(41,249,109,181)(42,248,110,180)(43,247,111,179)(44,246,112,178)(45,245,113,177)(46,244,114,176)(47,243,115,175)(48,242,116,174)(49,241,117,173)(50,240,118,172)(51,239,119,171)(52,238,120,170)(53,237,121,169)(54,236,122,168)(55,235,123,167)(56,234,124,166)(57,233,125,165)(58,232,126,164)(59,231,127,163)(60,230,128,162)(61,229,129,161)(62,228,130,160)(63,227,131,159)(64,226,132,158)(65,225,133,157)(66,224,134,156)(67,223,135,155)(68,222,136,154)>;
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), (1,153,69,221)(2,152,70,220)(3,151,71,219)(4,150,72,218)(5,149,73,217)(6,148,74,216)(7,147,75,215)(8,146,76,214)(9,145,77,213)(10,144,78,212)(11,143,79,211)(12,142,80,210)(13,141,81,209)(14,140,82,208)(15,139,83,207)(16,138,84,206)(17,137,85,205)(18,272,86,204)(19,271,87,203)(20,270,88,202)(21,269,89,201)(22,268,90,200)(23,267,91,199)(24,266,92,198)(25,265,93,197)(26,264,94,196)(27,263,95,195)(28,262,96,194)(29,261,97,193)(30,260,98,192)(31,259,99,191)(32,258,100,190)(33,257,101,189)(34,256,102,188)(35,255,103,187)(36,254,104,186)(37,253,105,185)(38,252,106,184)(39,251,107,183)(40,250,108,182)(41,249,109,181)(42,248,110,180)(43,247,111,179)(44,246,112,178)(45,245,113,177)(46,244,114,176)(47,243,115,175)(48,242,116,174)(49,241,117,173)(50,240,118,172)(51,239,119,171)(52,238,120,170)(53,237,121,169)(54,236,122,168)(55,235,123,167)(56,234,124,166)(57,233,125,165)(58,232,126,164)(59,231,127,163)(60,230,128,162)(61,229,129,161)(62,228,130,160)(63,227,131,159)(64,226,132,158)(65,225,133,157)(66,224,134,156)(67,223,135,155)(68,222,136,154) );
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)], [(1,153,69,221),(2,152,70,220),(3,151,71,219),(4,150,72,218),(5,149,73,217),(6,148,74,216),(7,147,75,215),(8,146,76,214),(9,145,77,213),(10,144,78,212),(11,143,79,211),(12,142,80,210),(13,141,81,209),(14,140,82,208),(15,139,83,207),(16,138,84,206),(17,137,85,205),(18,272,86,204),(19,271,87,203),(20,270,88,202),(21,269,89,201),(22,268,90,200),(23,267,91,199),(24,266,92,198),(25,265,93,197),(26,264,94,196),(27,263,95,195),(28,262,96,194),(29,261,97,193),(30,260,98,192),(31,259,99,191),(32,258,100,190),(33,257,101,189),(34,256,102,188),(35,255,103,187),(36,254,104,186),(37,253,105,185),(38,252,106,184),(39,251,107,183),(40,250,108,182),(41,249,109,181),(42,248,110,180),(43,247,111,179),(44,246,112,178),(45,245,113,177),(46,244,114,176),(47,243,115,175),(48,242,116,174),(49,241,117,173),(50,240,118,172),(51,239,119,171),(52,238,120,170),(53,237,121,169),(54,236,122,168),(55,235,123,167),(56,234,124,166),(57,233,125,165),(58,232,126,164),(59,231,127,163),(60,230,128,162),(61,229,129,161),(62,228,130,160),(63,227,131,159),(64,226,132,158),(65,225,133,157),(66,224,134,156),(67,223,135,155),(68,222,136,154)]])
71 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 17A | ··· | 17H | 34A | ··· | 34H | 68A | ··· | 68P | 136A | ··· | 136AF |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 2 | 68 | 68 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
71 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D17 | D34 | D68 | Dic68 |
| kernel | Dic68 | C136 | Dic34 | C34 | C17 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 8 | 8 | 16 | 32 |
Matrix representation of Dic68 ►in GL2(𝔽137) generated by
| 134 | 17 |
| 120 | 96 |
| 130 | 88 |
| 15 | 7 |
G:=sub<GL(2,GF(137))| [134,120,17,96],[130,15,88,7] >;
Dic68 in GAP, Magma, Sage, TeX
{\rm Dic}_{68} % in TeX
G:=Group("Dic68"); // GroupNames label
G:=SmallGroup(272,8);
// by ID
G=gap.SmallGroup(272,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,40,61,66,182,42,6404]);
// Polycyclic
G:=Group<a,b|a^136=1,b^2=a^68,b*a*b^-1=a^-1>;
// generators/relations
Export