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G = D153order 306 = 2·32·17

Dihedral group

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

Aliases: D153, C9⋊D17, C17⋊D9, C3.D51, C1531C2, C51.1S3, sometimes denoted D306 or Dih153 or Dih306, SmallGroup(306,3)

Series: Derived Chief Lower central Upper central

C1C153 — D153
C1C3C51C153 — D153
C153 — D153
C1

Generators and relations for D153
 G = < a,b | a153=b2=1, bab=a-1 >

153C2
51S3
9D17
17D9
3D51

Smallest permutation representation of D153
On 153 points
Generators in S153
(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)
(2 153)(3 152)(4 151)(5 150)(6 149)(7 148)(8 147)(9 146)(10 145)(11 144)(12 143)(13 142)(14 141)(15 140)(16 139)(17 138)(18 137)(19 136)(20 135)(21 134)(22 133)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)

G:=sub<Sym(153)| (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), (2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)>;

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), (2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,138)(18,137)(19,136)(20,135)(21,134)(22,133)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78) );

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)], [(2,153),(3,152),(4,151),(5,150),(6,149),(7,148),(8,147),(9,146),(10,145),(11,144),(12,143),(13,142),(14,141),(15,140),(16,139),(17,138),(18,137),(19,136),(20,135),(21,134),(22,133),(23,132),(24,131),(25,130),(26,129),(27,128),(28,127),(29,126),(30,125),(31,124),(32,123),(33,122),(34,121),(35,120),(36,119),(37,118),(38,117),(39,116),(40,115),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78)])

78 conjugacy classes

class 1  2  3 9A9B9C17A···17H51A···51P153A···153AV
order12399917···1751···51153···153
size115322222···22···22···2

78 irreducible representations

dim1122222
type+++++++
imageC1C2S3D9D17D51D153
kernelD153C153C51C17C9C3C1
# reps111381648

Matrix representation of D153 in GL2(𝔽307) generated by

56271
3692
,
10
306306
G:=sub<GL(2,GF(307))| [56,36,271,92],[1,306,0,306] >;

D153 in GAP, Magma, Sage, TeX

D_{153}
% in TeX

G:=Group("D153");
// GroupNames label

G:=SmallGroup(306,3);
// by ID

G=gap.SmallGroup(306,3);
# by ID

G:=PCGroup([4,-2,-3,-17,-3,849,821,1154,3267]);
// Polycyclic

G:=Group<a,b|a^153=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D153 in TeX

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