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## G = D5×C34order 340 = 22·5·17

### Direct product of C34 and D5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C34
 Chief series C1 — C5 — C85 — D5×C17 — D5×C34
 Lower central C5 — D5×C34
 Upper central C1 — C34

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

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

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 5A 5B 10A 10B 17A ··· 17P 34A ··· 34P 34Q ··· 34AV 85A ··· 85AF 170A ··· 170AF order 1 2 2 2 5 5 10 10 17 ··· 17 34 ··· 34 34 ··· 34 85 ··· 85 170 ··· 170 size 1 1 5 5 2 2 2 2 1 ··· 1 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C17 C34 C34 D5 D10 D5×C17 D5×C34 kernel D5×C34 D5×C17 C170 D10 D5 C10 C34 C17 C2 C1 # reps 1 2 1 16 32 16 2 2 32 32

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

 500 0 0 500
,
 0 1 1020 457
,
 0 1020 1020 0
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

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