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G = D5×C34order 340 = 22·5·17

Direct product of C34 and D5

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

Aliases: D5×C34, C10⋊C34, C1703C2, C854C22, C5⋊(C2×C34), SmallGroup(340,13)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C34
C1C5C85D5×C17 — D5×C34
C5 — D5×C34
C1C34

Generators and relations for D5×C34
 G = < a,b,c | a34=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C34
5C34
5C2×C34

Smallest permutation representation of D5×C34
On 170 points
Generators in S170
(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)
(1 139 134 40 88)(2 140 135 41 89)(3 141 136 42 90)(4 142 103 43 91)(5 143 104 44 92)(6 144 105 45 93)(7 145 106 46 94)(8 146 107 47 95)(9 147 108 48 96)(10 148 109 49 97)(11 149 110 50 98)(12 150 111 51 99)(13 151 112 52 100)(14 152 113 53 101)(15 153 114 54 102)(16 154 115 55 69)(17 155 116 56 70)(18 156 117 57 71)(19 157 118 58 72)(20 158 119 59 73)(21 159 120 60 74)(22 160 121 61 75)(23 161 122 62 76)(24 162 123 63 77)(25 163 124 64 78)(26 164 125 65 79)(27 165 126 66 80)(28 166 127 67 81)(29 167 128 68 82)(30 168 129 35 83)(31 169 130 36 84)(32 170 131 37 85)(33 137 132 38 86)(34 138 133 39 87)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(33 69)(34 70)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 157)(42 158)(43 159)(44 160)(45 161)(46 162)(47 163)(48 164)(49 165)(50 166)(51 167)(52 168)(53 169)(54 170)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(103 120)(104 121)(105 122)(106 123)(107 124)(108 125)(109 126)(110 127)(111 128)(112 129)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)

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

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

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

136 conjugacy classes

class 1 2A2B2C5A5B10A10B17A···17P34A···34P34Q···34AV85A···85AF170A···170AF
order122255101017···1734···3434···3485···85170···170
size115522221···11···15···52···22···2

136 irreducible representations

dim1111112222
type+++++
imageC1C2C2C17C34C34D5D10D5×C17D5×C34
kernelD5×C34D5×C17C170D10D5C10C34C17C2C1
# reps121163216223232

Matrix representation of D5×C34 in GL2(𝔽1021) generated by

5000
0500
,
01
1020457
,
01020
10200
G:=sub<GL(2,GF(1021))| [500,0,0,500],[0,1020,1,457],[0,1020,1020,0] >;

D5×C34 in GAP, Magma, Sage, TeX

D_5\times C_{34}
% in TeX

G:=Group("D5xC34");
// GroupNames label

G:=SmallGroup(340,13);
// by ID

G=gap.SmallGroup(340,13);
# by ID

G:=PCGroup([4,-2,-2,-17,-5,4355]);
// Polycyclic

G:=Group<a,b,c|a^34=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C34 in TeX

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