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G = S3×C53order 318 = 2·3·53

Direct product of C53 and S3

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

Aliases: S3×C53, C3⋊C106, C1593C2, SmallGroup(318,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C53
C1C3C159 — S3×C53
C3 — S3×C53
C1C53

Generators and relations for S3×C53
 G = < a,b,c | a53=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C106

Smallest permutation representation of S3×C53
On 159 points
Generators in S159
(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)
(1 77 122)(2 78 123)(3 79 124)(4 80 125)(5 81 126)(6 82 127)(7 83 128)(8 84 129)(9 85 130)(10 86 131)(11 87 132)(12 88 133)(13 89 134)(14 90 135)(15 91 136)(16 92 137)(17 93 138)(18 94 139)(19 95 140)(20 96 141)(21 97 142)(22 98 143)(23 99 144)(24 100 145)(25 101 146)(26 102 147)(27 103 148)(28 104 149)(29 105 150)(30 106 151)(31 54 152)(32 55 153)(33 56 154)(34 57 155)(35 58 156)(36 59 157)(37 60 158)(38 61 159)(39 62 107)(40 63 108)(41 64 109)(42 65 110)(43 66 111)(44 67 112)(45 68 113)(46 69 114)(47 70 115)(48 71 116)(49 72 117)(50 73 118)(51 74 119)(52 75 120)(53 76 121)
(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 159)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 145)(101 146)(102 147)(103 148)(104 149)(105 150)(106 151)

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

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

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

159 conjugacy classes

class 1  2  3 53A···53AZ106A···106AZ159A···159AZ
order12353···53106···106159···159
size1321···13···32···2

159 irreducible representations

dim111122
type+++
imageC1C2C53C106S3S3×C53
kernelS3×C53C159S3C3C53C1
# reps115252152

Matrix representation of S3×C53 in GL2(𝔽3181) generated by

20270
02027
,
03180
13180
,
01
10
G:=sub<GL(2,GF(3181))| [2027,0,0,2027],[0,1,3180,3180],[0,1,1,0] >;

S3×C53 in GAP, Magma, Sage, TeX

S_3\times C_{53}
% in TeX

G:=Group("S3xC53");
// GroupNames label

G:=SmallGroup(318,1);
// by ID

G=gap.SmallGroup(318,1);
# by ID

G:=PCGroup([3,-2,-53,-3,1910]);
// Polycyclic

G:=Group<a,b,c|a^53=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C53 in TeX

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