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G = D172order 344 = 23·43

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D172, C4⋊D43, C431D4, C1721C2, D861C2, C2.4D86, C86.3C22, sometimes denoted D344 or Dih172 or Dih344, SmallGroup(344,5)

Series: Derived Chief Lower central Upper central

C1C86 — D172
C1C43C86D86 — D172
C43C86 — D172
C1C2C4

Generators and relations for D172
 G = < a,b | a172=b2=1, bab=a-1 >

86C2
86C2
43C22
43C22
2D43
2D43
43D4

Smallest permutation representation of D172
On 172 points
Generators in S172
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172)
(1 172)(2 171)(3 170)(4 169)(5 168)(6 167)(7 166)(8 165)(9 164)(10 163)(11 162)(12 161)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(21 152)(22 151)(23 150)(24 149)(25 148)(26 147)(27 146)(28 145)(29 144)(30 143)(31 142)(32 141)(33 140)(34 139)(35 138)(36 137)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 121)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)(61 112)(62 111)(63 110)(64 109)(65 108)(66 107)(67 106)(68 105)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)(79 94)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)

G:=sub<Sym(172)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,172)(2,171)(3,170)(4,169)(5,168)(6,167)(7,166)(8,165)(9,164)(10,163)(11,162)(12,161)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,172)(2,171)(3,170)(4,169)(5,168)(6,167)(7,166)(8,165)(9,164)(10,163)(11,162)(12,161)(13,160)(14,159)(15,158)(16,157)(17,156)(18,155)(19,154)(20,153)(21,152)(22,151)(23,150)(24,149)(25,148)(26,147)(27,146)(28,145)(29,144)(30,143)(31,142)(32,141)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172)], [(1,172),(2,171),(3,170),(4,169),(5,168),(6,167),(7,166),(8,165),(9,164),(10,163),(11,162),(12,161),(13,160),(14,159),(15,158),(16,157),(17,156),(18,155),(19,154),(20,153),(21,152),(22,151),(23,150),(24,149),(25,148),(26,147),(27,146),(28,145),(29,144),(30,143),(31,142),(32,141),(33,140),(34,139),(35,138),(36,137),(37,136),(38,135),(39,134),(40,133),(41,132),(42,131),(43,130),(44,129),(45,128),(46,127),(47,126),(48,125),(49,124),(50,123),(51,122),(52,121),(53,120),(54,119),(55,118),(56,117),(57,116),(58,115),(59,114),(60,113),(61,112),(62,111),(63,110),(64,109),(65,108),(66,107),(67,106),(68,105),(69,104),(70,103),(71,102),(72,101),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95),(79,94),(80,93),(81,92),(82,91),(83,90),(84,89),(85,88),(86,87)])

89 conjugacy classes

class 1 2A2B2C 4 43A···43U86A···86U172A···172AP
order1222443···4386···86172···172
size11868622···22···22···2

89 irreducible representations

dim1112222
type+++++++
imageC1C2C2D4D43D86D172
kernelD172C172D86C43C4C2C1
# reps1121212142

Matrix representation of D172 in GL2(𝔽173) generated by

11250
163167
,
123111
7150
G:=sub<GL(2,GF(173))| [112,163,50,167],[123,71,111,50] >;

D172 in GAP, Magma, Sage, TeX

D_{172}
% in TeX

G:=Group("D172");
// GroupNames label

G:=SmallGroup(344,5);
// by ID

G=gap.SmallGroup(344,5);
# by ID

G:=PCGroup([4,-2,-2,-2,-43,49,21,5379]);
// Polycyclic

G:=Group<a,b|a^172=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D172 in TeX

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