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G = D142order 284 = 22·71

Dihedral group

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

Aliases: D142, C2×D71, C142⋊C2, C71⋊C22, sometimes denoted D284 or Dih142 or Dih284, SmallGroup(284,3)

Series: Derived Chief Lower central Upper central

C1C71 — D142
C1C71D71 — D142
C71 — D142
C1C2

Generators and relations for D142
 G = < a,b | a142=b2=1, bab=a-1 >

71C2
71C2
71C22

Smallest permutation representation of D142
On 142 points
Generators in S142
(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)
(1 142)(2 141)(3 140)(4 139)(5 138)(6 137)(7 136)(8 135)(9 134)(10 133)(11 132)(12 131)(13 130)(14 129)(15 128)(16 127)(17 126)(18 125)(19 124)(20 123)(21 122)(22 121)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 96)(48 95)(49 94)(50 93)(51 92)(52 91)(53 90)(54 89)(55 88)(56 87)(57 86)(58 85)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)

G:=sub<Sym(142)| (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), (1,142)(2,141)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)>;

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), (1,142)(2,141)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72) );

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)], [(1,142),(2,141),(3,140),(4,139),(5,138),(6,137),(7,136),(8,135),(9,134),(10,133),(11,132),(12,131),(13,130),(14,129),(15,128),(16,127),(17,126),(18,125),(19,124),(20,123),(21,122),(22,121),(23,120),(24,119),(25,118),(26,117),(27,116),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(41,102),(42,101),(43,100),(44,99),(45,98),(46,97),(47,96),(48,95),(49,94),(50,93),(51,92),(52,91),(53,90),(54,89),(55,88),(56,87),(57,86),(58,85),(59,84),(60,83),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72)])

74 conjugacy classes

class 1 2A2B2C71A···71AI142A···142AI
order122271···71142···142
size1171712···22···2

74 irreducible representations

dim11122
type+++++
imageC1C2C2D71D142
kernelD142D71C142C2C1
# reps1213535

Matrix representation of D142 in GL3(𝔽569) generated by

56800
0236176
02360
,
100
0116412
0430453
G:=sub<GL(3,GF(569))| [568,0,0,0,236,236,0,176,0],[1,0,0,0,116,430,0,412,453] >;

D142 in GAP, Magma, Sage, TeX

D_{142}
% in TeX

G:=Group("D142");
// GroupNames label

G:=SmallGroup(284,3);
// by ID

G=gap.SmallGroup(284,3);
# by ID

G:=PCGroup([3,-2,-2,-71,2522]);
// Polycyclic

G:=Group<a,b|a^142=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D142 in TeX

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