metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D220, C4⋊D55, C55⋊4D4, C5⋊1D44, C44⋊1D5, C11⋊1D20, C220⋊1C2, C20⋊1D11, D110⋊1C2, C2.4D110, C22.10D10, C10.10D22, C110.10C22, sometimes denoted D440 or Dih220 or Dih440, SmallGroup(440,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D220
G = < a,b | a220=b2=1, bab=a-1 >
Smallest permutation representation of D220
►On 220 points
Generators in S220
(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)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(56 220)(57 219)(58 218)(59 217)(60 216)(61 215)(62 214)(63 213)(64 212)(65 211)(66 210)(67 209)(68 208)(69 207)(70 206)(71 205)(72 204)(73 203)(74 202)(75 201)(76 200)(77 199)(78 198)(79 197)(80 196)(81 195)(82 194)(83 193)(84 192)(85 191)(86 190)(87 189)(88 188)(89 187)(90 186)(91 185)(92 184)(93 183)(94 182)(95 181)(96 180)(97 179)(98 178)(99 177)(100 176)(101 175)(102 174)(103 173)(104 172)(105 171)(106 170)(107 169)(108 168)(109 167)(110 166)(111 165)(112 164)(113 163)(114 162)(115 161)(116 160)(117 159)(118 158)(119 157)(120 156)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 140)(137 139)
G:=sub<Sym(220)| (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(56,220)(57,219)(58,218)(59,217)(60,216)(61,215)(62,214)(63,213)(64,212)(65,211)(66,210)(67,209)(68,208)(69,207)(70,206)(71,205)(72,204)(73,203)(74,202)(75,201)(76,200)(77,199)(78,198)(79,197)(80,196)(81,195)(82,194)(83,193)(84,192)(85,191)(86,190)(87,189)(88,188)(89,187)(90,186)(91,185)(92,184)(93,183)(94,182)(95,181)(96,180)(97,179)(98,178)(99,177)(100,176)(101,175)(102,174)(103,173)(104,172)(105,171)(106,170)(107,169)(108,168)(109,167)(110,166)(111,165)(112,164)(113,163)(114,162)(115,161)(116,160)(117,159)(118,158)(119,157)(120,156)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139)>;
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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(56,220)(57,219)(58,218)(59,217)(60,216)(61,215)(62,214)(63,213)(64,212)(65,211)(66,210)(67,209)(68,208)(69,207)(70,206)(71,205)(72,204)(73,203)(74,202)(75,201)(76,200)(77,199)(78,198)(79,197)(80,196)(81,195)(82,194)(83,193)(84,192)(85,191)(86,190)(87,189)(88,188)(89,187)(90,186)(91,185)(92,184)(93,183)(94,182)(95,181)(96,180)(97,179)(98,178)(99,177)(100,176)(101,175)(102,174)(103,173)(104,172)(105,171)(106,170)(107,169)(108,168)(109,167)(110,166)(111,165)(112,164)(113,163)(114,162)(115,161)(116,160)(117,159)(118,158)(119,157)(120,156)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,140)(137,139) );
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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(56,220),(57,219),(58,218),(59,217),(60,216),(61,215),(62,214),(63,213),(64,212),(65,211),(66,210),(67,209),(68,208),(69,207),(70,206),(71,205),(72,204),(73,203),(74,202),(75,201),(76,200),(77,199),(78,198),(79,197),(80,196),(81,195),(82,194),(83,193),(84,192),(85,191),(86,190),(87,189),(88,188),(89,187),(90,186),(91,185),(92,184),(93,183),(94,182),(95,181),(96,180),(97,179),(98,178),(99,177),(100,176),(101,175),(102,174),(103,173),(104,172),(105,171),(106,170),(107,169),(108,168),(109,167),(110,166),(111,165),(112,164),(113,163),(114,162),(115,161),(116,160),(117,159),(118,158),(119,157),(120,156),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,140),(137,139)]])
113 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 10A | 10B | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 110 | 110 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
113 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D5 | D10 | D11 | D20 | D22 | D44 | D55 | D110 | D220 |
| kernel | D220 | C220 | D110 | C55 | C44 | C22 | C20 | C11 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of D220 ►in GL2(𝔽661) generated by
| 615 | 175 |
| 329 | 329 |
| 505 | 12 |
| 561 | 156 |
G:=sub<GL(2,GF(661))| [615,329,175,329],[505,561,12,156] >;
D220 in GAP, Magma, Sage, TeX
D_{220} % in TeX
G:=Group("D220"); // GroupNames label
G:=SmallGroup(440,36);
// by ID
G=gap.SmallGroup(440,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,61,26,643,10004]);
// Polycyclic
G:=Group<a,b|a^220=b^2=1,b*a*b=a^-1>;
// generators/relations
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