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G = D150order 300 = 22·3·52

Dihedral group

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

Aliases: D150, C2×D75, C50⋊S3, C6⋊D25, C32D50, C252D6, C5.D30, C1501C2, C752C22, C30.2D5, C10.2D15, C15.2D10, sometimes denoted D300 or Dih150 or Dih300, SmallGroup(300,11)

Series: Derived Chief Lower central Upper central

C1C75 — D150
C1C5C25C75D75 — D150
C75 — D150
C1C2

Generators and relations for D150
 G = < a,b | a150=b2=1, bab=a-1 >

75C2
75C2
75C22
25S3
25S3
15D5
15D5
25D6
15D10
5D15
5D15
3D25
3D25
5D30
3D50

Smallest permutation representation of D150
On 150 points
Generators in S150
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)

G:=sub<Sym(150)| (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)>;

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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76) );

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)], [(1,150),(2,149),(3,148),(4,147),(5,146),(6,145),(7,144),(8,143),(9,142),(10,141),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,114),(38,113),(39,112),(40,111),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,100),(52,99),(53,98),(54,97),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76)])

78 conjugacy classes

class 1 2A2B2C 3 5A5B 6 10A10B15A15B15C15D25A···25J30A30B30C30D50A···50J75A···75T150A···150T
order1222355610101515151525···253030303050···5075···75150···150
size11757522222222222···222222···22···22···2

78 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D5D6D10D15D25D30D50D75D150
kernelD150D75C150C50C30C25C15C10C6C5C3C2C1
# reps12112124104102020

Matrix representation of D150 in GL3(𝔽151) generated by

15000
03239
011228
,
100
03239
0148119
G:=sub<GL(3,GF(151))| [150,0,0,0,32,112,0,39,28],[1,0,0,0,32,148,0,39,119] >;

D150 in GAP, Magma, Sage, TeX

D_{150}
% in TeX

G:=Group("D150");
// GroupNames label

G:=SmallGroup(300,11);
// by ID

G=gap.SmallGroup(300,11);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,122,2163,418,6004]);
// Polycyclic

G:=Group<a,b|a^150=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D150 in TeX

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