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G = D226order 452 = 22·113

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D226, C2×D113, C226⋊C2, C113⋊C22, sometimes denoted D452 or Dih226 or Dih452, SmallGroup(452,4)

Series: Derived Chief Lower central Upper central

C1C113 — D226
C1C113D113 — D226
C113 — D226
C1C2

Generators and relations for D226
 G = < a,b | a226=b2=1, bab=a-1 >

113C2
113C2
113C22

Smallest permutation representation of D226
On 226 points
Generators in S226
(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)
(1 226)(2 225)(3 224)(4 223)(5 222)(6 221)(7 220)(8 219)(9 218)(10 217)(11 216)(12 215)(13 214)(14 213)(15 212)(16 211)(17 210)(18 209)(19 208)(20 207)(21 206)(22 205)(23 204)(24 203)(25 202)(26 201)(27 200)(28 199)(29 198)(30 197)(31 196)(32 195)(33 194)(34 193)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 186)(42 185)(43 184)(44 183)(45 182)(46 181)(47 180)(48 179)(49 178)(50 177)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 170)(58 169)(59 168)(60 167)(61 166)(62 165)(63 164)(64 163)(65 162)(66 161)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)(81 146)(82 145)(83 144)(84 143)(85 142)(86 141)(87 140)(88 139)(89 138)(90 137)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 130)(98 129)(99 128)(100 127)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)

G:=sub<Sym(226)| (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), (1,226)(2,225)(3,224)(4,223)(5,222)(6,221)(7,220)(8,219)(9,218)(10,217)(11,216)(12,215)(13,214)(14,213)(15,212)(16,211)(17,210)(18,209)(19,208)(20,207)(21,206)(22,205)(23,204)(24,203)(25,202)(26,201)(27,200)(28,199)(29,198)(30,197)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,170)(58,169)(59,168)(60,167)(61,166)(62,165)(63,164)(64,163)(65,162)(66,161)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(81,146)(82,145)(83,144)(84,143)(85,142)(86,141)(87,140)(88,139)(89,138)(90,137)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)>;

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), (1,226)(2,225)(3,224)(4,223)(5,222)(6,221)(7,220)(8,219)(9,218)(10,217)(11,216)(12,215)(13,214)(14,213)(15,212)(16,211)(17,210)(18,209)(19,208)(20,207)(21,206)(22,205)(23,204)(24,203)(25,202)(26,201)(27,200)(28,199)(29,198)(30,197)(31,196)(32,195)(33,194)(34,193)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,186)(42,185)(43,184)(44,183)(45,182)(46,181)(47,180)(48,179)(49,178)(50,177)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,170)(58,169)(59,168)(60,167)(61,166)(62,165)(63,164)(64,163)(65,162)(66,161)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)(81,146)(82,145)(83,144)(84,143)(85,142)(86,141)(87,140)(88,139)(89,138)(90,137)(91,136)(92,135)(93,134)(94,133)(95,132)(96,131)(97,130)(98,129)(99,128)(100,127)(101,126)(102,125)(103,124)(104,123)(105,122)(106,121)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114) );

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)], [(1,226),(2,225),(3,224),(4,223),(5,222),(6,221),(7,220),(8,219),(9,218),(10,217),(11,216),(12,215),(13,214),(14,213),(15,212),(16,211),(17,210),(18,209),(19,208),(20,207),(21,206),(22,205),(23,204),(24,203),(25,202),(26,201),(27,200),(28,199),(29,198),(30,197),(31,196),(32,195),(33,194),(34,193),(35,192),(36,191),(37,190),(38,189),(39,188),(40,187),(41,186),(42,185),(43,184),(44,183),(45,182),(46,181),(47,180),(48,179),(49,178),(50,177),(51,176),(52,175),(53,174),(54,173),(55,172),(56,171),(57,170),(58,169),(59,168),(60,167),(61,166),(62,165),(63,164),(64,163),(65,162),(66,161),(67,160),(68,159),(69,158),(70,157),(71,156),(72,155),(73,154),(74,153),(75,152),(76,151),(77,150),(78,149),(79,148),(80,147),(81,146),(82,145),(83,144),(84,143),(85,142),(86,141),(87,140),(88,139),(89,138),(90,137),(91,136),(92,135),(93,134),(94,133),(95,132),(96,131),(97,130),(98,129),(99,128),(100,127),(101,126),(102,125),(103,124),(104,123),(105,122),(106,121),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114)])

116 conjugacy classes

class 1 2A2B2C113A···113BD226A···226BD
order1222113···113226···226
size111131132···22···2

116 irreducible representations

dim11122
type+++++
imageC1C2C2D113D226
kernelD226D113C226C2C1
# reps1215656

Matrix representation of D226 in GL3(𝔽227) generated by

22600
018925
08156
,
100
090101
0203137
G:=sub<GL(3,GF(227))| [226,0,0,0,189,8,0,25,156],[1,0,0,0,90,203,0,101,137] >;

D226 in GAP, Magma, Sage, TeX

D_{226}
% in TeX

G:=Group("D226");
// GroupNames label

G:=SmallGroup(452,4);
// by ID

G=gap.SmallGroup(452,4);
# by ID

G:=PCGroup([3,-2,-2,-113,4034]);
// Polycyclic

G:=Group<a,b|a^226=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D226 in TeX

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