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G = D217order 434 = 2·7·31

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D217, C31⋊D7, C7⋊D31, C2171C2, sometimes denoted D434 or Dih217 or Dih434, SmallGroup(434,3)

Series: Derived Chief Lower central Upper central

C1C217 — D217
C1C31C217 — D217
C217 — D217
C1

Generators and relations for D217
 G = < a,b | a217=b2=1, bab=a-1 >

217C2
31D7
7D31

Smallest permutation representation of D217
On 217 points
Generators in S217
(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)
(1 217)(2 216)(3 215)(4 214)(5 213)(6 212)(7 211)(8 210)(9 209)(10 208)(11 207)(12 206)(13 205)(14 204)(15 203)(16 202)(17 201)(18 200)(19 199)(20 198)(21 197)(22 196)(23 195)(24 194)(25 193)(26 192)(27 191)(28 190)(29 189)(30 188)(31 187)(32 186)(33 185)(34 184)(35 183)(36 182)(37 181)(38 180)(39 179)(40 178)(41 177)(42 176)(43 175)(44 174)(45 173)(46 172)(47 171)(48 170)(49 169)(50 168)(51 167)(52 166)(53 165)(54 164)(55 163)(56 162)(57 161)(58 160)(59 159)(60 158)(61 157)(62 156)(63 155)(64 154)(65 153)(66 152)(67 151)(68 150)(69 149)(70 148)(71 147)(72 146)(73 145)(74 144)(75 143)(76 142)(77 141)(78 140)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)

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

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

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

110 conjugacy classes

class 1  2 7A7B7C31A···31O217A···217CL
order1277731···31217···217
size12172222···22···2

110 irreducible representations

dim11222
type+++++
imageC1C2D7D31D217
kernelD217C217C31C7C1
# reps1131590

Matrix representation of D217 in GL2(𝔽1303) generated by

1196385
1279817
,
32529
132978
G:=sub<GL(2,GF(1303))| [1196,1279,385,817],[325,132,29,978] >;

D217 in GAP, Magma, Sage, TeX

D_{217}
% in TeX

G:=Group("D217");
// GroupNames label

G:=SmallGroup(434,3);
// by ID

G=gap.SmallGroup(434,3);
# by ID

G:=PCGroup([3,-2,-7,-31,73,3782]);
// Polycyclic

G:=Group<a,b|a^217=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D217 in TeX

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