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G = C10×D17order 340 = 22·5·17

Direct product of C10 and D17

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

Aliases: C10×D17, C34⋊C10, C1702C2, C853C22, C17⋊(C2×C10), SmallGroup(340,12)

Series: Derived Chief Lower central Upper central

C1C17 — C10×D17
C1C17C85C5×D17 — C10×D17
C17 — C10×D17
C1C10

Generators and relations for C10×D17
 G = < a,b,c | a10=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C2
17C22
17C10
17C10
17C2×C10

Smallest permutation representation of C10×D17
On 170 points
Generators in S170
(1 122 79 104 68 90 39 160 31 146)(2 123 80 105 52 91 40 161 32 147)(3 124 81 106 53 92 41 162 33 148)(4 125 82 107 54 93 42 163 34 149)(5 126 83 108 55 94 43 164 18 150)(6 127 84 109 56 95 44 165 19 151)(7 128 85 110 57 96 45 166 20 152)(8 129 69 111 58 97 46 167 21 153)(9 130 70 112 59 98 47 168 22 137)(10 131 71 113 60 99 48 169 23 138)(11 132 72 114 61 100 49 170 24 139)(12 133 73 115 62 101 50 154 25 140)(13 134 74 116 63 102 51 155 26 141)(14 135 75 117 64 86 35 156 27 142)(15 136 76 118 65 87 36 157 28 143)(16 120 77 119 66 88 37 158 29 144)(17 121 78 103 67 89 38 159 30 145)
(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 89)(2 88)(3 87)(4 86)(5 102)(6 101)(7 100)(8 99)(9 98)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 90)(18 116)(19 115)(20 114)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 119)(33 118)(34 117)(35 125)(36 124)(37 123)(38 122)(39 121)(40 120)(41 136)(42 135)(43 134)(44 133)(45 132)(46 131)(47 130)(48 129)(49 128)(50 127)(51 126)(52 144)(53 143)(54 142)(55 141)(56 140)(57 139)(58 138)(59 137)(60 153)(61 152)(62 151)(63 150)(64 149)(65 148)(66 147)(67 146)(68 145)(69 169)(70 168)(71 167)(72 166)(73 165)(74 164)(75 163)(76 162)(77 161)(78 160)(79 159)(80 158)(81 157)(82 156)(83 155)(84 154)(85 170)

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

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

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

100 conjugacy classes

class 1 2A2B2C5A5B5C5D10A10B10C10D10E···10L17A···17H34A···34H85A···85AF170A···170AF
order122255551010101010···1017···1734···3485···85170···170
size1117171111111117···172···22···22···22···2

100 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D17D34C5×D17C10×D17
kernelC10×D17C5×D17C170D34D17C34C10C5C2C1
# reps121484883232

Matrix representation of C10×D17 in GL2(𝔽1021) generated by

3450
0345
,
01
1020560
,
01020
10200
G:=sub<GL(2,GF(1021))| [345,0,0,345],[0,1020,1,560],[0,1020,1020,0] >;

C10×D17 in GAP, Magma, Sage, TeX

C_{10}\times D_{17}
% in TeX

G:=Group("C10xD17");
// GroupNames label

G:=SmallGroup(340,12);
// by ID

G=gap.SmallGroup(340,12);
# by ID

G:=PCGroup([4,-2,-2,-5,-17,5123]);
// Polycyclic

G:=Group<a,b,c|a^10=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C10×D17 in TeX

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