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G = D140order 280 = 23·5·7

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D140, C4⋊D35, C354D4, C71D20, C51D28, C281D5, C201D7, C1401C2, D701C2, C2.4D70, C10.10D14, C14.10D10, C70.10C22, sometimes denoted D280 or Dih140 or Dih280, SmallGroup(280,26)

Series: Derived Chief Lower central Upper central

C1C70 — D140
C1C7C35C70D70 — D140
C35C70 — D140
C1C2C4

Generators and relations for D140
 G = < a,b | a140=b2=1, bab=a-1 >

70C2
70C2
35C22
35C22
14D5
14D5
10D7
10D7
35D4
7D10
7D10
5D14
5D14
2D35
2D35
7D20
5D28

Smallest permutation representation of D140
On 140 points
Generators in S140
(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)
(1 140)(2 139)(3 138)(4 137)(5 136)(6 135)(7 134)(8 133)(9 132)(10 131)(11 130)(12 129)(13 128)(14 127)(15 126)(16 125)(17 124)(18 123)(19 122)(20 121)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 100)(42 99)(43 98)(44 97)(45 96)(46 95)(47 94)(48 93)(49 92)(50 91)(51 90)(52 89)(53 88)(54 87)(55 86)(56 85)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)

G:=sub<Sym(140)| (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), (1,140)(2,139)(3,138)(4,137)(5,136)(6,135)(7,134)(8,133)(9,132)(10,131)(11,130)(12,129)(13,128)(14,127)(15,126)(16,125)(17,124)(18,123)(19,122)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)>;

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), (1,140)(2,139)(3,138)(4,137)(5,136)(6,135)(7,134)(8,133)(9,132)(10,131)(11,130)(12,129)(13,128)(14,127)(15,126)(16,125)(17,124)(18,123)(19,122)(20,121)(21,120)(22,119)(23,118)(24,117)(25,116)(26,115)(27,114)(28,113)(29,112)(30,111)(31,110)(32,109)(33,108)(34,107)(35,106)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71) );

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)], [(1,140),(2,139),(3,138),(4,137),(5,136),(6,135),(7,134),(8,133),(9,132),(10,131),(11,130),(12,129),(13,128),(14,127),(15,126),(16,125),(17,124),(18,123),(19,122),(20,121),(21,120),(22,119),(23,118),(24,117),(25,116),(26,115),(27,114),(28,113),(29,112),(30,111),(31,110),(32,109),(33,108),(34,107),(35,106),(36,105),(37,104),(38,103),(39,102),(40,101),(41,100),(42,99),(43,98),(44,97),(45,96),(46,95),(47,94),(48,93),(49,92),(50,91),(51,90),(52,89),(53,88),(54,87),(55,86),(56,85),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71)])

73 conjugacy classes

class 1 2A2B2C 4 5A5B7A7B7C10A10B14A14B14C20A20B20C20D28A···28F35A···35L70A···70L140A···140X
order122245577710101414142020202028···2835···3570···70140···140
size1170702222222222222222···22···22···22···2

73 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2D4D5D7D10D14D20D28D35D70D140
kernelD140C140D70C35C28C20C14C10C7C5C4C2C1
# reps1121232346121224

Matrix representation of D140 in GL2(𝔽281) generated by

245111
17030
,
245111
8236
G:=sub<GL(2,GF(281))| [245,170,111,30],[245,82,111,36] >;

D140 in GAP, Magma, Sage, TeX

D_{140}
% in TeX

G:=Group("D140");
// GroupNames label

G:=SmallGroup(280,26);
// by ID

G=gap.SmallGroup(280,26);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-7,61,26,643,6004]);
// Polycyclic

G:=Group<a,b|a^140=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D140 in TeX

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