metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic63, C9⋊Dic7, C7⋊Dic9, C63⋊1C4, C18.D7, C14.D9, C2.D63, C42.1S3, C6.1D21, C3.Dic21, C126.1C2, C21.1Dic3, SmallGroup(252,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C63 — Dic63 |
Generators and relations for Dic63
G = < a,b | a126=1, b2=a63, bab-1=a-1 >
Smallest permutation representation of Dic63
►Regular action on 252 points
Generators in S252
(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)
(1 148 64 211)(2 147 65 210)(3 146 66 209)(4 145 67 208)(5 144 68 207)(6 143 69 206)(7 142 70 205)(8 141 71 204)(9 140 72 203)(10 139 73 202)(11 138 74 201)(12 137 75 200)(13 136 76 199)(14 135 77 198)(15 134 78 197)(16 133 79 196)(17 132 80 195)(18 131 81 194)(19 130 82 193)(20 129 83 192)(21 128 84 191)(22 127 85 190)(23 252 86 189)(24 251 87 188)(25 250 88 187)(26 249 89 186)(27 248 90 185)(28 247 91 184)(29 246 92 183)(30 245 93 182)(31 244 94 181)(32 243 95 180)(33 242 96 179)(34 241 97 178)(35 240 98 177)(36 239 99 176)(37 238 100 175)(38 237 101 174)(39 236 102 173)(40 235 103 172)(41 234 104 171)(42 233 105 170)(43 232 106 169)(44 231 107 168)(45 230 108 167)(46 229 109 166)(47 228 110 165)(48 227 111 164)(49 226 112 163)(50 225 113 162)(51 224 114 161)(52 223 115 160)(53 222 116 159)(54 221 117 158)(55 220 118 157)(56 219 119 156)(57 218 120 155)(58 217 121 154)(59 216 122 153)(60 215 123 152)(61 214 124 151)(62 213 125 150)(63 212 126 149)
G:=sub<Sym(252)| (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), (1,148,64,211)(2,147,65,210)(3,146,66,209)(4,145,67,208)(5,144,68,207)(6,143,69,206)(7,142,70,205)(8,141,71,204)(9,140,72,203)(10,139,73,202)(11,138,74,201)(12,137,75,200)(13,136,76,199)(14,135,77,198)(15,134,78,197)(16,133,79,196)(17,132,80,195)(18,131,81,194)(19,130,82,193)(20,129,83,192)(21,128,84,191)(22,127,85,190)(23,252,86,189)(24,251,87,188)(25,250,88,187)(26,249,89,186)(27,248,90,185)(28,247,91,184)(29,246,92,183)(30,245,93,182)(31,244,94,181)(32,243,95,180)(33,242,96,179)(34,241,97,178)(35,240,98,177)(36,239,99,176)(37,238,100,175)(38,237,101,174)(39,236,102,173)(40,235,103,172)(41,234,104,171)(42,233,105,170)(43,232,106,169)(44,231,107,168)(45,230,108,167)(46,229,109,166)(47,228,110,165)(48,227,111,164)(49,226,112,163)(50,225,113,162)(51,224,114,161)(52,223,115,160)(53,222,116,159)(54,221,117,158)(55,220,118,157)(56,219,119,156)(57,218,120,155)(58,217,121,154)(59,216,122,153)(60,215,123,152)(61,214,124,151)(62,213,125,150)(63,212,126,149)>;
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), (1,148,64,211)(2,147,65,210)(3,146,66,209)(4,145,67,208)(5,144,68,207)(6,143,69,206)(7,142,70,205)(8,141,71,204)(9,140,72,203)(10,139,73,202)(11,138,74,201)(12,137,75,200)(13,136,76,199)(14,135,77,198)(15,134,78,197)(16,133,79,196)(17,132,80,195)(18,131,81,194)(19,130,82,193)(20,129,83,192)(21,128,84,191)(22,127,85,190)(23,252,86,189)(24,251,87,188)(25,250,88,187)(26,249,89,186)(27,248,90,185)(28,247,91,184)(29,246,92,183)(30,245,93,182)(31,244,94,181)(32,243,95,180)(33,242,96,179)(34,241,97,178)(35,240,98,177)(36,239,99,176)(37,238,100,175)(38,237,101,174)(39,236,102,173)(40,235,103,172)(41,234,104,171)(42,233,105,170)(43,232,106,169)(44,231,107,168)(45,230,108,167)(46,229,109,166)(47,228,110,165)(48,227,111,164)(49,226,112,163)(50,225,113,162)(51,224,114,161)(52,223,115,160)(53,222,116,159)(54,221,117,158)(55,220,118,157)(56,219,119,156)(57,218,120,155)(58,217,121,154)(59,216,122,153)(60,215,123,152)(61,214,124,151)(62,213,125,150)(63,212,126,149) );
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)], [(1,148,64,211),(2,147,65,210),(3,146,66,209),(4,145,67,208),(5,144,68,207),(6,143,69,206),(7,142,70,205),(8,141,71,204),(9,140,72,203),(10,139,73,202),(11,138,74,201),(12,137,75,200),(13,136,76,199),(14,135,77,198),(15,134,78,197),(16,133,79,196),(17,132,80,195),(18,131,81,194),(19,130,82,193),(20,129,83,192),(21,128,84,191),(22,127,85,190),(23,252,86,189),(24,251,87,188),(25,250,88,187),(26,249,89,186),(27,248,90,185),(28,247,91,184),(29,246,92,183),(30,245,93,182),(31,244,94,181),(32,243,95,180),(33,242,96,179),(34,241,97,178),(35,240,98,177),(36,239,99,176),(37,238,100,175),(38,237,101,174),(39,236,102,173),(40,235,103,172),(41,234,104,171),(42,233,105,170),(43,232,106,169),(44,231,107,168),(45,230,108,167),(46,229,109,166),(47,228,110,165),(48,227,111,164),(49,226,112,163),(50,225,113,162),(51,224,114,161),(52,223,115,160),(53,222,116,159),(54,221,117,158),(55,220,118,157),(56,219,119,156),(57,218,120,155),(58,217,121,154),(59,216,122,153),(60,215,123,152),(61,214,124,151),(62,213,125,150),(63,212,126,149)]])
66 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 9A | 9B | 9C | 14A | 14B | 14C | 18A | 18B | 18C | 21A | ··· | 21F | 42A | ··· | 42F | 63A | ··· | 63R | 126A | ··· | 126R |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 9 | 9 | 9 | 14 | 14 | 14 | 18 | 18 | 18 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 2 | 63 | 63 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D7 | D9 | Dic7 | Dic9 | D21 | Dic21 | D63 | Dic63 |
| kernel | Dic63 | C126 | C63 | C42 | C21 | C18 | C14 | C9 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 18 | 18 |
Matrix representation of Dic63 ►in GL3(𝔽757) generated by
| 756 | 0 | 0 |
| 0 | 167 | 311 |
| 0 | 446 | 613 |
| 87 | 0 | 0 |
| 0 | 342 | 600 |
| 0 | 258 | 415 |
G:=sub<GL(3,GF(757))| [756,0,0,0,167,446,0,311,613],[87,0,0,0,342,258,0,600,415] >;
Dic63 in GAP, Magma, Sage, TeX
{\rm Dic}_{63} % in TeX
G:=Group("Dic63"); // GroupNames label
G:=SmallGroup(252,5);
// by ID
G=gap.SmallGroup(252,5);
# by ID
G:=PCGroup([5,-2,-2,-3,-7,-3,10,1382,642,1443,4204]);
// Polycyclic
G:=Group<a,b|a^126=1,b^2=a^63,b*a*b^-1=a^-1>;
// generators/relations
Export