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G = D9×C19order 342 = 2·32·19

Direct product of C19 and D9

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

Aliases: D9×C19, C9⋊C38, C1713C2, C57.2S3, C3.(S3×C19), SmallGroup(342,3)

Series: Derived Chief Lower central Upper central

C1C9 — D9×C19
C1C3C9C171 — D9×C19
C9 — D9×C19
C1C19

Generators and relations for D9×C19
 G = < a,b,c | a19=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

9C2
3S3
9C38
3S3×C19

Smallest permutation representation of D9×C19
On 171 points
Generators in S171
(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)
(1 84 44 66 23 116 148 97 156)(2 85 45 67 24 117 149 98 157)(3 86 46 68 25 118 150 99 158)(4 87 47 69 26 119 151 100 159)(5 88 48 70 27 120 152 101 160)(6 89 49 71 28 121 134 102 161)(7 90 50 72 29 122 135 103 162)(8 91 51 73 30 123 136 104 163)(9 92 52 74 31 124 137 105 164)(10 93 53 75 32 125 138 106 165)(11 94 54 76 33 126 139 107 166)(12 95 55 58 34 127 140 108 167)(13 77 56 59 35 128 141 109 168)(14 78 57 60 36 129 142 110 169)(15 79 39 61 37 130 143 111 170)(16 80 40 62 38 131 144 112 171)(17 81 41 63 20 132 145 113 153)(18 82 42 64 21 133 146 114 154)(19 83 43 65 22 115 147 96 155)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 169)(15 170)(16 171)(17 153)(18 154)(19 155)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 109)(78 110)(79 111)(80 112)(81 113)(82 114)(83 96)(84 97)(85 98)(86 99)(87 100)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)

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

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

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

114 conjugacy classes

class 1  2  3 9A9B9C19A···19R38A···38R57A···57R171A···171BB
order12399919···1938···3857···57171···171
size1922221···19···92···22···2

114 irreducible representations

dim11112222
type++++
imageC1C2C19C38S3D9S3×C19D9×C19
kernelD9×C19C171D9C9C57C19C3C1
# reps111818131854

Matrix representation of D9×C19 in GL2(𝔽2053) generated by

1490
0149
,
1647974
1079673
,
1079406
1380974
G:=sub<GL(2,GF(2053))| [149,0,0,149],[1647,1079,974,673],[1079,1380,406,974] >;

D9×C19 in GAP, Magma, Sage, TeX

D_9\times C_{19}
% in TeX

G:=Group("D9xC19");
// GroupNames label

G:=SmallGroup(342,3);
// by ID

G=gap.SmallGroup(342,3);
# by ID

G:=PCGroup([4,-2,-19,-3,-3,2282,82,3651]);
// Polycyclic

G:=Group<a,b,c|a^19=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D9×C19 in TeX

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