metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic66, C4.D33, C33⋊2Q8, C44.1S3, C6.8D22, C2.3D66, C22.8D6, C3⋊2Dic22, C11⋊2Dic6, C132.1C2, C12.1D11, C66.8C22, Dic33.1C2, SmallGroup(264,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic66
G = < a,b | a132=1, b2=a66, bab-1=a-1 >
Smallest permutation representation of Dic66
►Regular action on 264 points
Generators in S264
(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)
(1 244 67 178)(2 243 68 177)(3 242 69 176)(4 241 70 175)(5 240 71 174)(6 239 72 173)(7 238 73 172)(8 237 74 171)(9 236 75 170)(10 235 76 169)(11 234 77 168)(12 233 78 167)(13 232 79 166)(14 231 80 165)(15 230 81 164)(16 229 82 163)(17 228 83 162)(18 227 84 161)(19 226 85 160)(20 225 86 159)(21 224 87 158)(22 223 88 157)(23 222 89 156)(24 221 90 155)(25 220 91 154)(26 219 92 153)(27 218 93 152)(28 217 94 151)(29 216 95 150)(30 215 96 149)(31 214 97 148)(32 213 98 147)(33 212 99 146)(34 211 100 145)(35 210 101 144)(36 209 102 143)(37 208 103 142)(38 207 104 141)(39 206 105 140)(40 205 106 139)(41 204 107 138)(42 203 108 137)(43 202 109 136)(44 201 110 135)(45 200 111 134)(46 199 112 133)(47 198 113 264)(48 197 114 263)(49 196 115 262)(50 195 116 261)(51 194 117 260)(52 193 118 259)(53 192 119 258)(54 191 120 257)(55 190 121 256)(56 189 122 255)(57 188 123 254)(58 187 124 253)(59 186 125 252)(60 185 126 251)(61 184 127 250)(62 183 128 249)(63 182 129 248)(64 181 130 247)(65 180 131 246)(66 179 132 245)
G:=sub<Sym(264)| (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), (1,244,67,178)(2,243,68,177)(3,242,69,176)(4,241,70,175)(5,240,71,174)(6,239,72,173)(7,238,73,172)(8,237,74,171)(9,236,75,170)(10,235,76,169)(11,234,77,168)(12,233,78,167)(13,232,79,166)(14,231,80,165)(15,230,81,164)(16,229,82,163)(17,228,83,162)(18,227,84,161)(19,226,85,160)(20,225,86,159)(21,224,87,158)(22,223,88,157)(23,222,89,156)(24,221,90,155)(25,220,91,154)(26,219,92,153)(27,218,93,152)(28,217,94,151)(29,216,95,150)(30,215,96,149)(31,214,97,148)(32,213,98,147)(33,212,99,146)(34,211,100,145)(35,210,101,144)(36,209,102,143)(37,208,103,142)(38,207,104,141)(39,206,105,140)(40,205,106,139)(41,204,107,138)(42,203,108,137)(43,202,109,136)(44,201,110,135)(45,200,111,134)(46,199,112,133)(47,198,113,264)(48,197,114,263)(49,196,115,262)(50,195,116,261)(51,194,117,260)(52,193,118,259)(53,192,119,258)(54,191,120,257)(55,190,121,256)(56,189,122,255)(57,188,123,254)(58,187,124,253)(59,186,125,252)(60,185,126,251)(61,184,127,250)(62,183,128,249)(63,182,129,248)(64,181,130,247)(65,180,131,246)(66,179,132,245)>;
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), (1,244,67,178)(2,243,68,177)(3,242,69,176)(4,241,70,175)(5,240,71,174)(6,239,72,173)(7,238,73,172)(8,237,74,171)(9,236,75,170)(10,235,76,169)(11,234,77,168)(12,233,78,167)(13,232,79,166)(14,231,80,165)(15,230,81,164)(16,229,82,163)(17,228,83,162)(18,227,84,161)(19,226,85,160)(20,225,86,159)(21,224,87,158)(22,223,88,157)(23,222,89,156)(24,221,90,155)(25,220,91,154)(26,219,92,153)(27,218,93,152)(28,217,94,151)(29,216,95,150)(30,215,96,149)(31,214,97,148)(32,213,98,147)(33,212,99,146)(34,211,100,145)(35,210,101,144)(36,209,102,143)(37,208,103,142)(38,207,104,141)(39,206,105,140)(40,205,106,139)(41,204,107,138)(42,203,108,137)(43,202,109,136)(44,201,110,135)(45,200,111,134)(46,199,112,133)(47,198,113,264)(48,197,114,263)(49,196,115,262)(50,195,116,261)(51,194,117,260)(52,193,118,259)(53,192,119,258)(54,191,120,257)(55,190,121,256)(56,189,122,255)(57,188,123,254)(58,187,124,253)(59,186,125,252)(60,185,126,251)(61,184,127,250)(62,183,128,249)(63,182,129,248)(64,181,130,247)(65,180,131,246)(66,179,132,245) );
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)], [(1,244,67,178),(2,243,68,177),(3,242,69,176),(4,241,70,175),(5,240,71,174),(6,239,72,173),(7,238,73,172),(8,237,74,171),(9,236,75,170),(10,235,76,169),(11,234,77,168),(12,233,78,167),(13,232,79,166),(14,231,80,165),(15,230,81,164),(16,229,82,163),(17,228,83,162),(18,227,84,161),(19,226,85,160),(20,225,86,159),(21,224,87,158),(22,223,88,157),(23,222,89,156),(24,221,90,155),(25,220,91,154),(26,219,92,153),(27,218,93,152),(28,217,94,151),(29,216,95,150),(30,215,96,149),(31,214,97,148),(32,213,98,147),(33,212,99,146),(34,211,100,145),(35,210,101,144),(36,209,102,143),(37,208,103,142),(38,207,104,141),(39,206,105,140),(40,205,106,139),(41,204,107,138),(42,203,108,137),(43,202,109,136),(44,201,110,135),(45,200,111,134),(46,199,112,133),(47,198,113,264),(48,197,114,263),(49,196,115,262),(50,195,116,261),(51,194,117,260),(52,193,118,259),(53,192,119,258),(54,191,120,257),(55,190,121,256),(56,189,122,255),(57,188,123,254),(58,187,124,253),(59,186,125,252),(60,185,126,251),(61,184,127,250),(62,183,128,249),(63,182,129,248),(64,181,130,247),(65,180,131,246),(66,179,132,245)]])
69 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 11A | ··· | 11E | 12A | 12B | 22A | ··· | 22E | 33A | ··· | 33J | 44A | ··· | 44J | 66A | ··· | 66J | 132A | ··· | 132T |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 11 | ··· | 11 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 2 | 2 | 66 | 66 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
69 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D6 | D11 | Dic6 | D22 | D33 | Dic22 | D66 | Dic66 |
| kernel | Dic66 | Dic33 | C132 | C44 | C33 | C22 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 5 | 2 | 5 | 10 | 10 | 10 | 20 |
Matrix representation of Dic66 ►in GL2(𝔽397) generated by
| 189 | 49 |
| 122 | 290 |
| 95 | 386 |
| 207 | 302 |
G:=sub<GL(2,GF(397))| [189,122,49,290],[95,207,386,302] >;
Dic66 in GAP, Magma, Sage, TeX
{\rm Dic}_{66} % in TeX
G:=Group("Dic66"); // GroupNames label
G:=SmallGroup(264,23);
// by ID
G=gap.SmallGroup(264,23);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-11,20,61,26,323,6004]);
// Polycyclic
G:=Group<a,b|a^132=1,b^2=a^66,b*a*b^-1=a^-1>;
// generators/relations
Export