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G = S3×C58order 348 = 22·3·29

Direct product of C58 and S3

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

Aliases: S3×C58, C6⋊C58, C1743C2, C874C22, C3⋊(C2×C58), SmallGroup(348,10)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C58
C1C3C87S3×C29 — S3×C58
C3 — S3×C58
C1C58

Generators and relations for S3×C58
 G = < a,b,c | a58=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C58
3C58
3C2×C58

Smallest permutation representation of S3×C58
On 174 points
Generators in S174
(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)
(1 139 73)(2 140 74)(3 141 75)(4 142 76)(5 143 77)(6 144 78)(7 145 79)(8 146 80)(9 147 81)(10 148 82)(11 149 83)(12 150 84)(13 151 85)(14 152 86)(15 153 87)(16 154 88)(17 155 89)(18 156 90)(19 157 91)(20 158 92)(21 159 93)(22 160 94)(23 161 95)(24 162 96)(25 163 97)(26 164 98)(27 165 99)(28 166 100)(29 167 101)(30 168 102)(31 169 103)(32 170 104)(33 171 105)(34 172 106)(35 173 107)(36 174 108)(37 117 109)(38 118 110)(39 119 111)(40 120 112)(41 121 113)(42 122 114)(43 123 115)(44 124 116)(45 125 59)(46 126 60)(47 127 61)(48 128 62)(49 129 63)(50 130 64)(51 131 65)(52 132 66)(53 133 67)(54 134 68)(55 135 69)(56 136 70)(57 137 71)(58 138 72)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(59 154)(60 155)(61 156)(62 157)(63 158)(64 159)(65 160)(66 161)(67 162)(68 163)(69 164)(70 165)(71 166)(72 167)(73 168)(74 169)(75 170)(76 171)(77 172)(78 173)(79 174)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)(89 126)(90 127)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 136)(100 137)(101 138)(102 139)(103 140)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)

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

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

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

174 conjugacy classes

class 1 2A2B2C 3  6 29A···29AB58A···58AB58AC···58CF87A···87AB174A···174AB
order12223629···2958···5858···5887···87174···174
size1133221···11···13···32···22···2

174 irreducible representations

dim1111112222
type+++++
imageC1C2C2C29C58C58S3D6S3×C29S3×C58
kernelS3×C58S3×C29C174D6S3C6C58C29C2C1
# reps121285628112828

Matrix representation of S3×C58 in GL2(𝔽349) generated by

1810
0181
,
0348
1348
,
0348
3480
G:=sub<GL(2,GF(349))| [181,0,0,181],[0,1,348,348],[0,348,348,0] >;

S3×C58 in GAP, Magma, Sage, TeX

S_3\times C_{58}
% in TeX

G:=Group("S3xC58");
// GroupNames label

G:=SmallGroup(348,10);
// by ID

G=gap.SmallGroup(348,10);
# by ID

G:=PCGroup([4,-2,-2,-29,-3,3715]);
// Polycyclic

G:=Group<a,b,c|a^58=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×C58 in TeX

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