metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic60, C8.D15, C15⋊4Q16, C40.1S3, C2.5D60, C6.3D20, C24.1D5, C5⋊1Dic12, C3⋊1Dic20, C120.1C2, C10.3D12, C4.10D30, C30.21D4, C20.45D6, C12.45D10, C60.52C22, Dic30.1C2, SmallGroup(240,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic60
G = < a,b | a120=1, b2=a60, bab-1=a-1 >
Smallest permutation representation of Dic60
►Regular action on 240 points
Generators in S240
(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)
(1 181 61 121)(2 180 62 240)(3 179 63 239)(4 178 64 238)(5 177 65 237)(6 176 66 236)(7 175 67 235)(8 174 68 234)(9 173 69 233)(10 172 70 232)(11 171 71 231)(12 170 72 230)(13 169 73 229)(14 168 74 228)(15 167 75 227)(16 166 76 226)(17 165 77 225)(18 164 78 224)(19 163 79 223)(20 162 80 222)(21 161 81 221)(22 160 82 220)(23 159 83 219)(24 158 84 218)(25 157 85 217)(26 156 86 216)(27 155 87 215)(28 154 88 214)(29 153 89 213)(30 152 90 212)(31 151 91 211)(32 150 92 210)(33 149 93 209)(34 148 94 208)(35 147 95 207)(36 146 96 206)(37 145 97 205)(38 144 98 204)(39 143 99 203)(40 142 100 202)(41 141 101 201)(42 140 102 200)(43 139 103 199)(44 138 104 198)(45 137 105 197)(46 136 106 196)(47 135 107 195)(48 134 108 194)(49 133 109 193)(50 132 110 192)(51 131 111 191)(52 130 112 190)(53 129 113 189)(54 128 114 188)(55 127 115 187)(56 126 116 186)(57 125 117 185)(58 124 118 184)(59 123 119 183)(60 122 120 182)
G:=sub<Sym(240)| (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), (1,181,61,121)(2,180,62,240)(3,179,63,239)(4,178,64,238)(5,177,65,237)(6,176,66,236)(7,175,67,235)(8,174,68,234)(9,173,69,233)(10,172,70,232)(11,171,71,231)(12,170,72,230)(13,169,73,229)(14,168,74,228)(15,167,75,227)(16,166,76,226)(17,165,77,225)(18,164,78,224)(19,163,79,223)(20,162,80,222)(21,161,81,221)(22,160,82,220)(23,159,83,219)(24,158,84,218)(25,157,85,217)(26,156,86,216)(27,155,87,215)(28,154,88,214)(29,153,89,213)(30,152,90,212)(31,151,91,211)(32,150,92,210)(33,149,93,209)(34,148,94,208)(35,147,95,207)(36,146,96,206)(37,145,97,205)(38,144,98,204)(39,143,99,203)(40,142,100,202)(41,141,101,201)(42,140,102,200)(43,139,103,199)(44,138,104,198)(45,137,105,197)(46,136,106,196)(47,135,107,195)(48,134,108,194)(49,133,109,193)(50,132,110,192)(51,131,111,191)(52,130,112,190)(53,129,113,189)(54,128,114,188)(55,127,115,187)(56,126,116,186)(57,125,117,185)(58,124,118,184)(59,123,119,183)(60,122,120,182)>;
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), (1,181,61,121)(2,180,62,240)(3,179,63,239)(4,178,64,238)(5,177,65,237)(6,176,66,236)(7,175,67,235)(8,174,68,234)(9,173,69,233)(10,172,70,232)(11,171,71,231)(12,170,72,230)(13,169,73,229)(14,168,74,228)(15,167,75,227)(16,166,76,226)(17,165,77,225)(18,164,78,224)(19,163,79,223)(20,162,80,222)(21,161,81,221)(22,160,82,220)(23,159,83,219)(24,158,84,218)(25,157,85,217)(26,156,86,216)(27,155,87,215)(28,154,88,214)(29,153,89,213)(30,152,90,212)(31,151,91,211)(32,150,92,210)(33,149,93,209)(34,148,94,208)(35,147,95,207)(36,146,96,206)(37,145,97,205)(38,144,98,204)(39,143,99,203)(40,142,100,202)(41,141,101,201)(42,140,102,200)(43,139,103,199)(44,138,104,198)(45,137,105,197)(46,136,106,196)(47,135,107,195)(48,134,108,194)(49,133,109,193)(50,132,110,192)(51,131,111,191)(52,130,112,190)(53,129,113,189)(54,128,114,188)(55,127,115,187)(56,126,116,186)(57,125,117,185)(58,124,118,184)(59,123,119,183)(60,122,120,182) );
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)], [(1,181,61,121),(2,180,62,240),(3,179,63,239),(4,178,64,238),(5,177,65,237),(6,176,66,236),(7,175,67,235),(8,174,68,234),(9,173,69,233),(10,172,70,232),(11,171,71,231),(12,170,72,230),(13,169,73,229),(14,168,74,228),(15,167,75,227),(16,166,76,226),(17,165,77,225),(18,164,78,224),(19,163,79,223),(20,162,80,222),(21,161,81,221),(22,160,82,220),(23,159,83,219),(24,158,84,218),(25,157,85,217),(26,156,86,216),(27,155,87,215),(28,154,88,214),(29,153,89,213),(30,152,90,212),(31,151,91,211),(32,150,92,210),(33,149,93,209),(34,148,94,208),(35,147,95,207),(36,146,96,206),(37,145,97,205),(38,144,98,204),(39,143,99,203),(40,142,100,202),(41,141,101,201),(42,140,102,200),(43,139,103,199),(44,138,104,198),(45,137,105,197),(46,136,106,196),(47,135,107,195),(48,134,108,194),(49,133,109,193),(50,132,110,192),(51,131,111,191),(52,130,112,190),(53,129,113,189),(54,128,114,188),(55,127,115,187),(56,126,116,186),(57,125,117,185),(58,124,118,184),(59,123,119,183),(60,122,120,182)])
Dic60 is a maximal subgroup of
D40.S3 D24.D5 C3⋊Dic40 C5⋊Dic24 C48⋊D5 Dic120 D8.D15 C15⋊7Q32 D5×Dic12 Dic60⋊C2 S3×Dic20 D24⋊7D5 D40⋊7S3 C40.2D6 C40.69D6 C8.D30 D8⋊3D15 SD16⋊D15 Q16×D15
Dic60 is a maximal quotient of
Dic30⋊8C4 C120⋊9C4
63 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | ··· | 60H | 120A | ··· | 120P |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D5 | D6 | Q16 | D10 | D12 | D15 | D20 | Dic12 | D30 | Dic20 | D60 | Dic60 |
| kernel | Dic60 | C120 | Dic30 | C40 | C30 | C24 | C20 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of Dic60 ►in GL2(𝔽241) generated by
| 187 | 202 |
| 39 | 46 |
| 86 | 52 |
| 159 | 155 |
G:=sub<GL(2,GF(241))| [187,39,202,46],[86,159,52,155] >;
Dic60 in GAP, Magma, Sage, TeX
{\rm Dic}_{60} % in TeX
G:=Group("Dic60"); // GroupNames label
G:=SmallGroup(240,69);
// by ID
G=gap.SmallGroup(240,69);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,79,218,50,964,6917]);
// Polycyclic
G:=Group<a,b|a^120=1,b^2=a^60,b*a*b^-1=a^-1>;
// generators/relations
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