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G = S3×C56order 336 = 24·3·7

Direct product of C56 and S3

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

Aliases: S3×C56, C244C14, C16812C2, D6.2C28, C28.56D6, C84.73C22, Dic3.2C28, C3⋊C86C14, C31(C2×C56), C216(C2×C8), C2.1(S3×C28), C6.1(C2×C28), (C4×S3).3C14, (S3×C28).6C2, (S3×C14).4C4, C4.12(S3×C14), C14.15(C4×S3), C42.24(C2×C4), C12.12(C2×C14), (C7×Dic3).4C4, (C7×C3⋊C8)⋊13C2, SmallGroup(336,74)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C56
C1C3C6C12C84S3×C28 — S3×C56
C3 — S3×C56
C1C56

Generators and relations for S3×C56
 G = < a,b,c | a56=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C4
3C14
3C14
3C8
3C2×C4
3C28
3C2×C14
3C2×C8
3C2×C28
3C56
3C2×C56

Smallest permutation representation of S3×C56
On 168 points
Generators in S168
(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)
(1 156 70)(2 157 71)(3 158 72)(4 159 73)(5 160 74)(6 161 75)(7 162 76)(8 163 77)(9 164 78)(10 165 79)(11 166 80)(12 167 81)(13 168 82)(14 113 83)(15 114 84)(16 115 85)(17 116 86)(18 117 87)(19 118 88)(20 119 89)(21 120 90)(22 121 91)(23 122 92)(24 123 93)(25 124 94)(26 125 95)(27 126 96)(28 127 97)(29 128 98)(30 129 99)(31 130 100)(32 131 101)(33 132 102)(34 133 103)(35 134 104)(36 135 105)(37 136 106)(38 137 107)(39 138 108)(40 139 109)(41 140 110)(42 141 111)(43 142 112)(44 143 57)(45 144 58)(46 145 59)(47 146 60)(48 147 61)(49 148 62)(50 149 63)(51 150 64)(52 151 65)(53 152 66)(54 153 67)(55 154 68)(56 155 69)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 161)(104 162)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 113)(112 114)

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

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

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

168 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 7A···7F8A8B8C8D8E8F8G8H12A12B14A···14F14G···14R21A···21F24A24B24C24D28A···28L28M···28X42A···42F56A···56X56Y···56AV84A···84L168A···168X
order12223444467···788888888121214···1414···1421···212424242428···2828···2842···4256···5656···5684···84168···168
size11332113321···111113333221···13···32···222221···13···32···21···13···32···22···2

168 irreducible representations

dim1111111111111122222222
type++++++
imageC1C2C2C2C4C4C7C8C14C14C14C28C28C56S3D6C4×S3S3×C7S3×C8S3×C14S3×C28S3×C56
kernelS3×C56C7×C3⋊C8C168S3×C28C7×Dic3S3×C14S3×C8S3×C7C3⋊C8C24C4×S3Dic3D6S3C56C28C14C8C7C4C2C1
# reps111122686661212481126461224

Matrix representation of S3×C56 in GL3(𝔽337) generated by

8500
0360
0036
,
100
0336336
010
,
100
001
010
G:=sub<GL(3,GF(337))| [85,0,0,0,36,0,0,0,36],[1,0,0,0,336,1,0,336,0],[1,0,0,0,0,1,0,1,0] >;

S3×C56 in GAP, Magma, Sage, TeX

S_3\times C_{56}
% in TeX

G:=Group("S3xC56");
// GroupNames label

G:=SmallGroup(336,74);
// by ID

G=gap.SmallGroup(336,74);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,175,69,8069]);
// Polycyclic

G:=Group<a,b,c|a^56=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×C56 in TeX

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