direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D210, C2×D105, C6⋊D35, C10⋊D21, C14⋊D15, C35⋊7D6, C5⋊2D42, C3⋊2D70, C7⋊2D30, C70⋊1S3, C42⋊1D5, C30⋊1D7, C21⋊7D10, C15⋊7D14, C210⋊1C2, C105⋊8C22, sometimes denoted D420 or Dih210 or Dih420, SmallGroup(420,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — D210 |
Generators and relations for D210
G = < a,b | a210=b2=1, bab=a-1 >
105C2
105C2
105C22
35S3
35S3
21D5
21D5
15D7
15D7
35D6
21D10
15D14
7D15
7D15
5D21
5D21
3D35
3D35
7D30
5D42
3D70
Smallest permutation representation of D210
►On 210 points
Generators in S210
(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)
(1 210)(2 209)(3 208)(4 207)(5 206)(6 205)(7 204)(8 203)(9 202)(10 201)(11 200)(12 199)(13 198)(14 197)(15 196)(16 195)(17 194)(18 193)(19 192)(20 191)(21 190)(22 189)(23 188)(24 187)(25 186)(26 185)(27 184)(28 183)(29 182)(30 181)(31 180)(32 179)(33 178)(34 177)(35 176)(36 175)(37 174)(38 173)(39 172)(40 171)(41 170)(42 169)(43 168)(44 167)(45 166)(46 165)(47 164)(48 163)(49 162)(50 161)(51 160)(52 159)(53 158)(54 157)(55 156)(56 155)(57 154)(58 153)(59 152)(60 151)(61 150)(62 149)(63 148)(64 147)(65 146)(66 145)(67 144)(68 143)(69 142)(70 141)(71 140)(72 139)(73 138)(74 137)(75 136)(76 135)(77 134)(78 133)(79 132)(80 131)(81 130)(82 129)(83 128)(84 127)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)
G:=sub<Sym(210)| (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), (1,210)(2,209)(3,208)(4,207)(5,206)(6,205)(7,204)(8,203)(9,202)(10,201)(11,200)(12,199)(13,198)(14,197)(15,196)(16,195)(17,194)(18,193)(19,192)(20,191)(21,190)(22,189)(23,188)(24,187)(25,186)(26,185)(27,184)(28,183)(29,182)(30,181)(31,180)(32,179)(33,178)(34,177)(35,176)(36,175)(37,174)(38,173)(39,172)(40,171)(41,170)(42,169)(43,168)(44,167)(45,166)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,150)(62,149)(63,148)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)>;
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), (1,210)(2,209)(3,208)(4,207)(5,206)(6,205)(7,204)(8,203)(9,202)(10,201)(11,200)(12,199)(13,198)(14,197)(15,196)(16,195)(17,194)(18,193)(19,192)(20,191)(21,190)(22,189)(23,188)(24,187)(25,186)(26,185)(27,184)(28,183)(29,182)(30,181)(31,180)(32,179)(33,178)(34,177)(35,176)(36,175)(37,174)(38,173)(39,172)(40,171)(41,170)(42,169)(43,168)(44,167)(45,166)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,154)(58,153)(59,152)(60,151)(61,150)(62,149)(63,148)(64,147)(65,146)(66,145)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,138)(74,137)(75,136)(76,135)(77,134)(78,133)(79,132)(80,131)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106) );
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)], [(1,210),(2,209),(3,208),(4,207),(5,206),(6,205),(7,204),(8,203),(9,202),(10,201),(11,200),(12,199),(13,198),(14,197),(15,196),(16,195),(17,194),(18,193),(19,192),(20,191),(21,190),(22,189),(23,188),(24,187),(25,186),(26,185),(27,184),(28,183),(29,182),(30,181),(31,180),(32,179),(33,178),(34,177),(35,176),(36,175),(37,174),(38,173),(39,172),(40,171),(41,170),(42,169),(43,168),(44,167),(45,166),(46,165),(47,164),(48,163),(49,162),(50,161),(51,160),(52,159),(53,158),(54,157),(55,156),(56,155),(57,154),(58,153),(59,152),(60,151),(61,150),(62,149),(63,148),(64,147),(65,146),(66,145),(67,144),(68,143),(69,142),(70,141),(71,140),(72,139),(73,138),(74,137),(75,136),(76,135),(77,134),(78,133),(79,132),(80,131),(81,130),(82,129),(83,128),(84,127),(85,126),(86,125),(87,124),(88,123),(89,122),(90,121),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 30A | 30B | 30C | 30D | 35A | ··· | 35L | 42A | ··· | 42F | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 30 | 30 | 30 | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 105 | 105 | 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 |
108 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 | D5 | D6 | D7 | D10 | D14 | D15 | D21 | D30 | D35 | D42 | D70 | D105 | D210 |
| kernel | D210 | D105 | C210 | C70 | C42 | C35 | C30 | C21 | C15 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 3 | 2 | 3 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of D210 ►in GL3(𝔽211) generated by
| 210 | 0 | 0 |
| 0 | 115 | 93 |
| 0 | 118 | 178 |
| 1 | 0 | 0 |
| 0 | 73 | 62 |
| 0 | 57 | 138 |
G:=sub<GL(3,GF(211))| [210,0,0,0,115,118,0,93,178],[1,0,0,0,73,57,0,62,138] >;
D210 in GAP, Magma, Sage, TeX
D_{210} % in TeX
G:=Group("D210"); // GroupNames label
G:=SmallGroup(420,40);
// by ID
G=gap.SmallGroup(420,40);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,122,963,9004]);
// Polycyclic
G:=Group<a,b|a^210=b^2=1,b*a*b=a^-1>;
// generators/relations
Export