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G = S3×C59order 354 = 2·3·59

Direct product of C59 and S3

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

Aliases: S3×C59, C3⋊C118, C1773C2, SmallGroup(354,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C59
C1C3C177 — S3×C59
C3 — S3×C59
C1C59

Generators and relations for S3×C59
 G = < a,b,c | a59=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C118

Smallest permutation representation of S3×C59
On 177 points
Generators in S177
(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)(119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177)
(1 112 156)(2 113 157)(3 114 158)(4 115 159)(5 116 160)(6 117 161)(7 118 162)(8 60 163)(9 61 164)(10 62 165)(11 63 166)(12 64 167)(13 65 168)(14 66 169)(15 67 170)(16 68 171)(17 69 172)(18 70 173)(19 71 174)(20 72 175)(21 73 176)(22 74 177)(23 75 119)(24 76 120)(25 77 121)(26 78 122)(27 79 123)(28 80 124)(29 81 125)(30 82 126)(31 83 127)(32 84 128)(33 85 129)(34 86 130)(35 87 131)(36 88 132)(37 89 133)(38 90 134)(39 91 135)(40 92 136)(41 93 137)(42 94 138)(43 95 139)(44 96 140)(45 97 141)(46 98 142)(47 99 143)(48 100 144)(49 101 145)(50 102 146)(51 103 147)(52 104 148)(53 105 149)(54 106 150)(55 107 151)(56 108 152)(57 109 153)(58 110 154)(59 111 155)
(60 163)(61 164)(62 165)(63 166)(64 167)(65 168)(66 169)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 176)(74 177)(75 119)(76 120)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 127)(84 128)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 144)(101 145)(102 146)(103 147)(104 148)(105 149)(106 150)(107 151)(108 152)(109 153)(110 154)(111 155)(112 156)(113 157)(114 158)(115 159)(116 160)(117 161)(118 162)

G:=sub<Sym(177)| (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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,112,156)(2,113,157)(3,114,158)(4,115,159)(5,116,160)(6,117,161)(7,118,162)(8,60,163)(9,61,164)(10,62,165)(11,63,166)(12,64,167)(13,65,168)(14,66,169)(15,67,170)(16,68,171)(17,69,172)(18,70,173)(19,71,174)(20,72,175)(21,73,176)(22,74,177)(23,75,119)(24,76,120)(25,77,121)(26,78,122)(27,79,123)(28,80,124)(29,81,125)(30,82,126)(31,83,127)(32,84,128)(33,85,129)(34,86,130)(35,87,131)(36,88,132)(37,89,133)(38,90,134)(39,91,135)(40,92,136)(41,93,137)(42,94,138)(43,95,139)(44,96,140)(45,97,141)(46,98,142)(47,99,143)(48,100,144)(49,101,145)(50,102,146)(51,103,147)(52,104,148)(53,105,149)(54,106,150)(55,107,151)(56,108,152)(57,109,153)(58,110,154)(59,111,155), (60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,177)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,144)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)>;

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)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177), (1,112,156)(2,113,157)(3,114,158)(4,115,159)(5,116,160)(6,117,161)(7,118,162)(8,60,163)(9,61,164)(10,62,165)(11,63,166)(12,64,167)(13,65,168)(14,66,169)(15,67,170)(16,68,171)(17,69,172)(18,70,173)(19,71,174)(20,72,175)(21,73,176)(22,74,177)(23,75,119)(24,76,120)(25,77,121)(26,78,122)(27,79,123)(28,80,124)(29,81,125)(30,82,126)(31,83,127)(32,84,128)(33,85,129)(34,86,130)(35,87,131)(36,88,132)(37,89,133)(38,90,134)(39,91,135)(40,92,136)(41,93,137)(42,94,138)(43,95,139)(44,96,140)(45,97,141)(46,98,142)(47,99,143)(48,100,144)(49,101,145)(50,102,146)(51,103,147)(52,104,148)(53,105,149)(54,106,150)(55,107,151)(56,108,152)(57,109,153)(58,110,154)(59,111,155), (60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,177)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,144)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162) );

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),(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177)], [(1,112,156),(2,113,157),(3,114,158),(4,115,159),(5,116,160),(6,117,161),(7,118,162),(8,60,163),(9,61,164),(10,62,165),(11,63,166),(12,64,167),(13,65,168),(14,66,169),(15,67,170),(16,68,171),(17,69,172),(18,70,173),(19,71,174),(20,72,175),(21,73,176),(22,74,177),(23,75,119),(24,76,120),(25,77,121),(26,78,122),(27,79,123),(28,80,124),(29,81,125),(30,82,126),(31,83,127),(32,84,128),(33,85,129),(34,86,130),(35,87,131),(36,88,132),(37,89,133),(38,90,134),(39,91,135),(40,92,136),(41,93,137),(42,94,138),(43,95,139),(44,96,140),(45,97,141),(46,98,142),(47,99,143),(48,100,144),(49,101,145),(50,102,146),(51,103,147),(52,104,148),(53,105,149),(54,106,150),(55,107,151),(56,108,152),(57,109,153),(58,110,154),(59,111,155)], [(60,163),(61,164),(62,165),(63,166),(64,167),(65,168),(66,169),(67,170),(68,171),(69,172),(70,173),(71,174),(72,175),(73,176),(74,177),(75,119),(76,120),(77,121),(78,122),(79,123),(80,124),(81,125),(82,126),(83,127),(84,128),(85,129),(86,130),(87,131),(88,132),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(97,141),(98,142),(99,143),(100,144),(101,145),(102,146),(103,147),(104,148),(105,149),(106,150),(107,151),(108,152),(109,153),(110,154),(111,155),(112,156),(113,157),(114,158),(115,159),(116,160),(117,161),(118,162)])

177 conjugacy classes

class 1  2  3 59A···59BF118A···118BF177A···177BF
order12359···59118···118177···177
size1321···13···32···2

177 irreducible representations

dim111122
type+++
imageC1C2C59C118S3S3×C59
kernelS3×C59C177S3C3C59C1
# reps115858158

Matrix representation of S3×C59 in GL2(𝔽709) generated by

6640
0664
,
0708
1708
,
01
10
G:=sub<GL(2,GF(709))| [664,0,0,664],[0,1,708,708],[0,1,1,0] >;

S3×C59 in GAP, Magma, Sage, TeX

S_3\times C_{59}
% in TeX

G:=Group("S3xC59");
// GroupNames label

G:=SmallGroup(354,1);
// by ID

G=gap.SmallGroup(354,1);
# by ID

G:=PCGroup([3,-2,-59,-3,2126]);
// Polycyclic

G:=Group<a,b,c|a^59=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×C59 in TeX

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