Copied to
clipboard

G = D175order 350 = 2·52·7

Dihedral group

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

Aliases: D175, C25⋊D7, C7⋊D25, C5.D35, C1751C2, C35.1D5, sometimes denoted D350 or Dih175 or Dih350, SmallGroup(350,3)

Series: Derived Chief Lower central Upper central

C1C175 — D175
C1C5C35C175 — D175
C175 — D175
C1

Generators and relations for D175
 G = < a,b | a175=b2=1, bab=a-1 >

175C2
35D5
25D7
7D25
5D35

Smallest permutation representation of D175
On 175 points
Generators in S175
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175)
(2 175)(3 174)(4 173)(5 172)(6 171)(7 170)(8 169)(9 168)(10 167)(11 166)(12 165)(13 164)(14 163)(15 162)(16 161)(17 160)(18 159)(19 158)(20 157)(21 156)(22 155)(23 154)(24 153)(25 152)(26 151)(27 150)(28 149)(29 148)(30 147)(31 146)(32 145)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 136)(42 135)(43 134)(44 133)(45 132)(46 131)(47 130)(48 129)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 100)(78 99)(79 98)(80 97)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)

G:=sub<Sym(175)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175), (2,175)(3,174)(4,173)(5,172)(6,171)(7,170)(8,169)(9,168)(10,167)(11,166)(12,165)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175), (2,175)(3,174)(4,173)(5,172)(6,171)(7,170)(8,169)(9,168)(10,167)(11,166)(12,165)(13,164)(14,163)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,154)(24,153)(25,152)(26,151)(27,150)(28,149)(29,148)(30,147)(31,146)(32,145)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,136)(42,135)(43,134)(44,133)(45,132)(46,131)(47,130)(48,129)(49,128)(50,127)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175)], [(2,175),(3,174),(4,173),(5,172),(6,171),(7,170),(8,169),(9,168),(10,167),(11,166),(12,165),(13,164),(14,163),(15,162),(16,161),(17,160),(18,159),(19,158),(20,157),(21,156),(22,155),(23,154),(24,153),(25,152),(26,151),(27,150),(28,149),(29,148),(30,147),(31,146),(32,145),(33,144),(34,143),(35,142),(36,141),(37,140),(38,139),(39,138),(40,137),(41,136),(42,135),(43,134),(44,133),(45,132),(46,131),(47,130),(48,129),(49,128),(50,127),(51,126),(52,125),(53,124),(54,123),(55,122),(56,121),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(73,104),(74,103),(75,102),(76,101),(77,100),(78,99),(79,98),(80,97),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89)])

89 conjugacy classes

class 1  2 5A5B7A7B7C25A···25J35A···35L175A···175BH
order125577725···2535···35175···175
size1175222222···22···22···2

89 irreducible representations

dim1122222
type+++++++
imageC1C2D5D7D25D35D175
kernelD175C175C35C25C7C5C1
# reps1123101260

Matrix representation of D175 in GL2(𝔽701) generated by

697280
421147
,
10
674700
G:=sub<GL(2,GF(701))| [697,421,280,147],[1,674,0,700] >;

D175 in GAP, Magma, Sage, TeX

D_{175}
% in TeX

G:=Group("D175");
// GroupNames label

G:=SmallGroup(350,3);
// by ID

G=gap.SmallGroup(350,3);
# by ID

G:=PCGroup([4,-2,-5,-7,-5,625,1125,722,4483]);
// Polycyclic

G:=Group<a,b|a^175=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D175 in TeX

׿
×
𝔽