metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D240, C5⋊1D48, C3⋊1D80, C80⋊1S3, C48⋊1D5, C15⋊4D16, C240⋊1C2, C16⋊1D15, C6.1D40, C4.1D60, D120⋊1C2, C30.22D8, C10.1D24, C40.65D6, C2.3D120, C8.13D30, C20.26D12, C12.26D20, C24.65D10, C60.157D4, C120.78C22, sometimes denoted D480 or Dih240 or Dih480, SmallGroup(480,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D240
G = < a,b | a240=b2=1, bab=a-1 >
120C2
120C2
60C22
60C22
40S3
40S3
24D5
24D5
30D4
30D4
20D6
20D6
12D10
12D10
8D15
8D15
15D8
15D8
10D12
10D12
6D20
6D20
4D30
4D30
15D16
5D24
5D24
3D40
3D40
2D60
2D60
5D48
3D80
Smallest permutation representation of D240
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 240)(17 239)(18 238)(19 237)(20 236)(21 235)(22 234)(23 233)(24 232)(25 231)(26 230)(27 229)(28 228)(29 227)(30 226)(31 225)(32 224)(33 223)(34 222)(35 221)(36 220)(37 219)(38 218)(39 217)(40 216)(41 215)(42 214)(43 213)(44 212)(45 211)(46 210)(47 209)(48 208)(49 207)(50 206)(51 205)(52 204)(53 203)(54 202)(55 201)(56 200)(57 199)(58 198)(59 197)(60 196)(61 195)(62 194)(63 193)(64 192)(65 191)(66 190)(67 189)(68 188)(69 187)(70 186)(71 185)(72 184)(73 183)(74 182)(75 181)(76 180)(77 179)(78 178)(79 177)(80 176)(81 175)(82 174)(83 173)(84 172)(85 171)(86 170)(87 169)(88 168)(89 167)(90 166)(91 165)(92 164)(93 163)(94 162)(95 161)(96 160)(97 159)(98 158)(99 157)(100 156)(101 155)(102 154)(103 153)(104 152)(105 151)(106 150)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(113 143)(114 142)(115 141)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,240)(17,239)(18,238)(19,237)(20,236)(21,235)(22,234)(23,233)(24,232)(25,231)(26,230)(27,229)(28,228)(29,227)(30,226)(31,225)(32,224)(33,223)(34,222)(35,221)(36,220)(37,219)(38,218)(39,217)(40,216)(41,215)(42,214)(43,213)(44,212)(45,211)(46,210)(47,209)(48,208)(49,207)(50,206)(51,205)(52,204)(53,203)(54,202)(55,201)(56,200)(57,199)(58,198)(59,197)(60,196)(61,195)(62,194)(63,193)(64,192)(65,191)(66,190)(67,189)(68,188)(69,187)(70,186)(71,185)(72,184)(73,183)(74,182)(75,181)(76,180)(77,179)(78,178)(79,177)(80,176)(81,175)(82,174)(83,173)(84,172)(85,171)(86,170)(87,169)(88,168)(89,167)(90,166)(91,165)(92,164)(93,163)(94,162)(95,161)(96,160)(97,159)(98,158)(99,157)(100,156)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,240)(17,239)(18,238)(19,237)(20,236)(21,235)(22,234)(23,233)(24,232)(25,231)(26,230)(27,229)(28,228)(29,227)(30,226)(31,225)(32,224)(33,223)(34,222)(35,221)(36,220)(37,219)(38,218)(39,217)(40,216)(41,215)(42,214)(43,213)(44,212)(45,211)(46,210)(47,209)(48,208)(49,207)(50,206)(51,205)(52,204)(53,203)(54,202)(55,201)(56,200)(57,199)(58,198)(59,197)(60,196)(61,195)(62,194)(63,193)(64,192)(65,191)(66,190)(67,189)(68,188)(69,187)(70,186)(71,185)(72,184)(73,183)(74,182)(75,181)(76,180)(77,179)(78,178)(79,177)(80,176)(81,175)(82,174)(83,173)(84,172)(85,171)(86,170)(87,169)(88,168)(89,167)(90,166)(91,165)(92,164)(93,163)(94,162)(95,161)(96,160)(97,159)(98,158)(99,157)(100,156)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,240),(17,239),(18,238),(19,237),(20,236),(21,235),(22,234),(23,233),(24,232),(25,231),(26,230),(27,229),(28,228),(29,227),(30,226),(31,225),(32,224),(33,223),(34,222),(35,221),(36,220),(37,219),(38,218),(39,217),(40,216),(41,215),(42,214),(43,213),(44,212),(45,211),(46,210),(47,209),(48,208),(49,207),(50,206),(51,205),(52,204),(53,203),(54,202),(55,201),(56,200),(57,199),(58,198),(59,197),(60,196),(61,195),(62,194),(63,193),(64,192),(65,191),(66,190),(67,189),(68,188),(69,187),(70,186),(71,185),(72,184),(73,183),(74,182),(75,181),(76,180),(77,179),(78,178),(79,177),(80,176),(81,175),(82,174),(83,173),(84,172),(85,171),(86,170),(87,169),(88,168),(89,167),(90,166),(91,165),(92,164),(93,163),(94,162),(95,161),(96,160),(97,159),(98,158),(99,157),(100,156),(101,155),(102,154),(103,153),(104,152),(105,151),(106,150),(107,149),(108,148),(109,147),(110,146),(111,145),(112,144),(113,143),(114,142),(115,141),(116,140),(117,139),(118,138),(119,137),(120,136),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129)]])
123 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 120 | 120 | 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 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
123 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D5 | D6 | D8 | D10 | D12 | D15 | D16 | D20 | D24 | D30 | D40 | D48 | D60 | D80 | D120 | D240 |
| kernel | D240 | C240 | D120 | C80 | C60 | C48 | C40 | C30 | C24 | C20 | C16 | 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 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of D240 ►in GL2(𝔽241) generated by
| 201 | 65 |
| 170 | 31 |
| 199 | 9 |
| 179 | 42 |
G:=sub<GL(2,GF(241))| [201,170,65,31],[199,179,9,42] >;
D240 in GAP, Magma, Sage, TeX
D_{240} % in TeX
G:=Group("D240"); // GroupNames label
G:=SmallGroup(480,159);
// by ID
G=gap.SmallGroup(480,159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,85,92,254,142,675,80,2693,18822]);
// Polycyclic
G:=Group<a,b|a^240=b^2=1,b*a*b=a^-1>;
// generators/relations
Export