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G = D144  order 288 = 25·32

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D144, C9⋊1D16, C3.D48, C16⋊1D9, C144⋊1C2, D72⋊1C2, C48.1S3, C18.1D8, C4.1D36, C6.1D24, C2.3D72, C24.67D6, C36.24D4, C8.13D18, C12.34D12, C72.14C22, sometimes denoted D288 or Dih144 or Dih288, SmallGroup(288,6)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C72 — D144
Chief series
C1 — C3 — C9 — C18 — C36 — C72 — D72 — D144
Lower central
C9 — C18 — C36 — C72 — D144
Upper central
C1 — C2 — C4 — C8 — C16

Generators and relations for D144
 G = < a,b | a144=b2=1, bab=a-1 >

C1
C2
72C2
72C2
C3
C4
36C22
36C22
C6
24S3
24S3
C9
C8
18D4
18D4
C12
12D6
12D6
C18
8D9
8D9
C16
9D8
9D8
C24
6D12
6D12
C36
4D18
4D18
9D16
C48
3D24
3D24
C72
2D36
2D36
3D48
C144
D72
D72
D144

Smallest permutation representation of D144
►On 144 points
Generators in S144
(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)

(1 144)(2 143)(3 142)(4 141)(5 140)(6 139)(7 138)(8 137)(9 136)(10 135)(11 134)(12 133)(13 132)(14 131)(15 130)(16 129)(17 128)(18 127)(19 126)(20 125)(21 124)(22 123)(23 122)(24 121)(25 120)(26 119)(27 118)(28 117)(29 116)(30 115)(31 114)(32 113)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)
 
G:=sub<Sym(144)| (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), (1,144)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)>;
 
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), (1,144)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,126)(20,125)(21,124)(22,123)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73) );
 
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)], [(1,144),(2,143),(3,142),(4,141),(5,140),(6,139),(7,138),(8,137),(9,136),(10,135),(11,134),(12,133),(13,132),(14,131),(15,130),(16,129),(17,128),(18,127),(19,126),(20,125),(21,124),(22,123),(23,122),(24,121),(25,120),(26,119),(27,118),(28,117),(29,116),(30,115),(31,114),(32,113),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(57,88),(58,87),(59,86),(60,85),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)]])
 

75 conjugacy classes

class 1 2A2B2C 3  4  6 8A8B9A9B9C12A12B16A16B16C16D18A18B18C24A24B24C24D36A···36F48A···48H72A···72L144A···144X
order1222346889991212161616161818182424242436···3648···4872···72144···144
size1172722222222222222222222222···22···22···22···2

75 irreducible representations

dim1112222222222222
type++++++++++++++++
imageC1C2C2S3D4D6D8D9D12D16D18D24D36D48D72D144
kernelD144C144D72C48C36C24C18C16C12C9C8C6C4C3C2C1
# reps112111232434681224

Matrix representation of D144 ►in GL2(𝔽433) generated by

27997
336376
,
336376
27997
G:=sub<GL(2,GF(433))| [279,336,97,376],[336,279,376,97] >;
 

D144 in GAP, Magma, Sage, TeX 

D_{144}
 
% in TeX
 
G:=Group("D144");
 
// GroupNames label
 
G:=SmallGroup(288,6);
 
// by ID
 
G=gap.SmallGroup(288,6);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,254,142,675,80,6725,292,9414]);
 
// Polycyclic
 
G:=Group<a,b|a^144=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D144 in TeX

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