metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic59, C59⋊C4, C2.D59, C118.C2, SmallGroup(236,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C59 — Dic59 |
Generators and relations for Dic59
G = < a,b | a118=1, b2=a59, bab-1=a-1 >
Smallest permutation representation of Dic59
►Regular action on 236 points
Generators in S236
(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)
(1 226 60 167)(2 225 61 166)(3 224 62 165)(4 223 63 164)(5 222 64 163)(6 221 65 162)(7 220 66 161)(8 219 67 160)(9 218 68 159)(10 217 69 158)(11 216 70 157)(12 215 71 156)(13 214 72 155)(14 213 73 154)(15 212 74 153)(16 211 75 152)(17 210 76 151)(18 209 77 150)(19 208 78 149)(20 207 79 148)(21 206 80 147)(22 205 81 146)(23 204 82 145)(24 203 83 144)(25 202 84 143)(26 201 85 142)(27 200 86 141)(28 199 87 140)(29 198 88 139)(30 197 89 138)(31 196 90 137)(32 195 91 136)(33 194 92 135)(34 193 93 134)(35 192 94 133)(36 191 95 132)(37 190 96 131)(38 189 97 130)(39 188 98 129)(40 187 99 128)(41 186 100 127)(42 185 101 126)(43 184 102 125)(44 183 103 124)(45 182 104 123)(46 181 105 122)(47 180 106 121)(48 179 107 120)(49 178 108 119)(50 177 109 236)(51 176 110 235)(52 175 111 234)(53 174 112 233)(54 173 113 232)(55 172 114 231)(56 171 115 230)(57 170 116 229)(58 169 117 228)(59 168 118 227)
G:=sub<Sym(236)| (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), (1,226,60,167)(2,225,61,166)(3,224,62,165)(4,223,63,164)(5,222,64,163)(6,221,65,162)(7,220,66,161)(8,219,67,160)(9,218,68,159)(10,217,69,158)(11,216,70,157)(12,215,71,156)(13,214,72,155)(14,213,73,154)(15,212,74,153)(16,211,75,152)(17,210,76,151)(18,209,77,150)(19,208,78,149)(20,207,79,148)(21,206,80,147)(22,205,81,146)(23,204,82,145)(24,203,83,144)(25,202,84,143)(26,201,85,142)(27,200,86,141)(28,199,87,140)(29,198,88,139)(30,197,89,138)(31,196,90,137)(32,195,91,136)(33,194,92,135)(34,193,93,134)(35,192,94,133)(36,191,95,132)(37,190,96,131)(38,189,97,130)(39,188,98,129)(40,187,99,128)(41,186,100,127)(42,185,101,126)(43,184,102,125)(44,183,103,124)(45,182,104,123)(46,181,105,122)(47,180,106,121)(48,179,107,120)(49,178,108,119)(50,177,109,236)(51,176,110,235)(52,175,111,234)(53,174,112,233)(54,173,113,232)(55,172,114,231)(56,171,115,230)(57,170,116,229)(58,169,117,228)(59,168,118,227)>;
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), (1,226,60,167)(2,225,61,166)(3,224,62,165)(4,223,63,164)(5,222,64,163)(6,221,65,162)(7,220,66,161)(8,219,67,160)(9,218,68,159)(10,217,69,158)(11,216,70,157)(12,215,71,156)(13,214,72,155)(14,213,73,154)(15,212,74,153)(16,211,75,152)(17,210,76,151)(18,209,77,150)(19,208,78,149)(20,207,79,148)(21,206,80,147)(22,205,81,146)(23,204,82,145)(24,203,83,144)(25,202,84,143)(26,201,85,142)(27,200,86,141)(28,199,87,140)(29,198,88,139)(30,197,89,138)(31,196,90,137)(32,195,91,136)(33,194,92,135)(34,193,93,134)(35,192,94,133)(36,191,95,132)(37,190,96,131)(38,189,97,130)(39,188,98,129)(40,187,99,128)(41,186,100,127)(42,185,101,126)(43,184,102,125)(44,183,103,124)(45,182,104,123)(46,181,105,122)(47,180,106,121)(48,179,107,120)(49,178,108,119)(50,177,109,236)(51,176,110,235)(52,175,111,234)(53,174,112,233)(54,173,113,232)(55,172,114,231)(56,171,115,230)(57,170,116,229)(58,169,117,228)(59,168,118,227) );
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)], [(1,226,60,167),(2,225,61,166),(3,224,62,165),(4,223,63,164),(5,222,64,163),(6,221,65,162),(7,220,66,161),(8,219,67,160),(9,218,68,159),(10,217,69,158),(11,216,70,157),(12,215,71,156),(13,214,72,155),(14,213,73,154),(15,212,74,153),(16,211,75,152),(17,210,76,151),(18,209,77,150),(19,208,78,149),(20,207,79,148),(21,206,80,147),(22,205,81,146),(23,204,82,145),(24,203,83,144),(25,202,84,143),(26,201,85,142),(27,200,86,141),(28,199,87,140),(29,198,88,139),(30,197,89,138),(31,196,90,137),(32,195,91,136),(33,194,92,135),(34,193,93,134),(35,192,94,133),(36,191,95,132),(37,190,96,131),(38,189,97,130),(39,188,98,129),(40,187,99,128),(41,186,100,127),(42,185,101,126),(43,184,102,125),(44,183,103,124),(45,182,104,123),(46,181,105,122),(47,180,106,121),(48,179,107,120),(49,178,108,119),(50,177,109,236),(51,176,110,235),(52,175,111,234),(53,174,112,233),(54,173,113,232),(55,172,114,231),(56,171,115,230),(57,170,116,229),(58,169,117,228),(59,168,118,227)]])
Dic59 is a maximal subgroup of
Dic118 C4×D59 C59⋊D4
Dic59 is a maximal quotient of C59⋊C8
62 conjugacy classes
| class | 1 | 2 | 4A | 4B | 59A | ··· | 59AC | 118A | ··· | 118AC |
| order | 1 | 2 | 4 | 4 | 59 | ··· | 59 | 118 | ··· | 118 |
| size | 1 | 1 | 59 | 59 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D59 | Dic59 |
| kernel | Dic59 | C118 | C59 | C2 | C1 |
| # reps | 1 | 1 | 2 | 29 | 29 |
Matrix representation of Dic59 ►in GL3(𝔽709) generated by
| 708 | 0 | 0 |
| 0 | 605 | 708 |
| 0 | 1 | 0 |
| 96 | 0 | 0 |
| 0 | 244 | 282 |
| 0 | 430 | 465 |
G:=sub<GL(3,GF(709))| [708,0,0,0,605,1,0,708,0],[96,0,0,0,244,430,0,282,465] >;
Dic59 in GAP, Magma, Sage, TeX
{\rm Dic}_{59} % in TeX
G:=Group("Dic59"); // GroupNames label
G:=SmallGroup(236,1);
// by ID
G=gap.SmallGroup(236,1);
# by ID
G:=PCGroup([3,-2,-2,-59,6,2090]);
// Polycyclic
G:=Group<a,b|a^118=1,b^2=a^59,b*a*b^-1=a^-1>;
// generators/relations
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