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G = S3×C62order 372 = 22·3·31

Direct product of C62 and S3

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

Aliases: S3×C62, C6⋊C62, C1863C2, C934C22, C3⋊(C2×C62), SmallGroup(372,13)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C62
C1C3C93S3×C31 — S3×C62
C3 — S3×C62
C1C62

Generators and relations for S3×C62
 G = < a,b,c | a62=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C62
3C62
3C2×C62

Smallest permutation representation of S3×C62
On 186 points
Generators in S186
(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)(125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186)
(1 164 88)(2 165 89)(3 166 90)(4 167 91)(5 168 92)(6 169 93)(7 170 94)(8 171 95)(9 172 96)(10 173 97)(11 174 98)(12 175 99)(13 176 100)(14 177 101)(15 178 102)(16 179 103)(17 180 104)(18 181 105)(19 182 106)(20 183 107)(21 184 108)(22 185 109)(23 186 110)(24 125 111)(25 126 112)(26 127 113)(27 128 114)(28 129 115)(29 130 116)(30 131 117)(31 132 118)(32 133 119)(33 134 120)(34 135 121)(35 136 122)(36 137 123)(37 138 124)(38 139 63)(39 140 64)(40 141 65)(41 142 66)(42 143 67)(43 144 68)(44 145 69)(45 146 70)(46 147 71)(47 148 72)(48 149 73)(49 150 74)(50 151 75)(51 152 76)(52 153 77)(53 154 78)(54 155 79)(55 156 80)(56 157 81)(57 158 82)(58 159 83)(59 160 84)(60 161 85)(61 162 86)(62 163 87)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(63 170)(64 171)(65 172)(66 173)(67 174)(68 175)(69 176)(70 177)(71 178)(72 179)(73 180)(74 181)(75 182)(76 183)(77 184)(78 185)(79 186)(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)(107 152)(108 153)(109 154)(110 155)(111 156)(112 157)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 169)

G:=sub<Sym(186)| (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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186), (1,164,88)(2,165,89)(3,166,90)(4,167,91)(5,168,92)(6,169,93)(7,170,94)(8,171,95)(9,172,96)(10,173,97)(11,174,98)(12,175,99)(13,176,100)(14,177,101)(15,178,102)(16,179,103)(17,180,104)(18,181,105)(19,182,106)(20,183,107)(21,184,108)(22,185,109)(23,186,110)(24,125,111)(25,126,112)(26,127,113)(27,128,114)(28,129,115)(29,130,116)(30,131,117)(31,132,118)(32,133,119)(33,134,120)(34,135,121)(35,136,122)(36,137,123)(37,138,124)(38,139,63)(39,140,64)(40,141,65)(41,142,66)(42,143,67)(43,144,68)(44,145,69)(45,146,70)(46,147,71)(47,148,72)(48,149,73)(49,150,74)(50,151,75)(51,152,76)(52,153,77)(53,154,78)(54,155,79)(55,156,80)(56,157,81)(57,158,82)(58,159,83)(59,160,84)(60,161,85)(61,162,86)(62,163,87), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,176)(70,177)(71,178)(72,179)(73,180)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(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)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,169)>;

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)(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186), (1,164,88)(2,165,89)(3,166,90)(4,167,91)(5,168,92)(6,169,93)(7,170,94)(8,171,95)(9,172,96)(10,173,97)(11,174,98)(12,175,99)(13,176,100)(14,177,101)(15,178,102)(16,179,103)(17,180,104)(18,181,105)(19,182,106)(20,183,107)(21,184,108)(22,185,109)(23,186,110)(24,125,111)(25,126,112)(26,127,113)(27,128,114)(28,129,115)(29,130,116)(30,131,117)(31,132,118)(32,133,119)(33,134,120)(34,135,121)(35,136,122)(36,137,123)(37,138,124)(38,139,63)(39,140,64)(40,141,65)(41,142,66)(42,143,67)(43,144,68)(44,145,69)(45,146,70)(46,147,71)(47,148,72)(48,149,73)(49,150,74)(50,151,75)(51,152,76)(52,153,77)(53,154,78)(54,155,79)(55,156,80)(56,157,81)(57,158,82)(58,159,83)(59,160,84)(60,161,85)(61,162,86)(62,163,87), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(63,170)(64,171)(65,172)(66,173)(67,174)(68,175)(69,176)(70,177)(71,178)(72,179)(73,180)(74,181)(75,182)(76,183)(77,184)(78,185)(79,186)(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)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,169) );

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),(125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186)], [(1,164,88),(2,165,89),(3,166,90),(4,167,91),(5,168,92),(6,169,93),(7,170,94),(8,171,95),(9,172,96),(10,173,97),(11,174,98),(12,175,99),(13,176,100),(14,177,101),(15,178,102),(16,179,103),(17,180,104),(18,181,105),(19,182,106),(20,183,107),(21,184,108),(22,185,109),(23,186,110),(24,125,111),(25,126,112),(26,127,113),(27,128,114),(28,129,115),(29,130,116),(30,131,117),(31,132,118),(32,133,119),(33,134,120),(34,135,121),(35,136,122),(36,137,123),(37,138,124),(38,139,63),(39,140,64),(40,141,65),(41,142,66),(42,143,67),(43,144,68),(44,145,69),(45,146,70),(46,147,71),(47,148,72),(48,149,73),(49,150,74),(50,151,75),(51,152,76),(52,153,77),(53,154,78),(54,155,79),(55,156,80),(56,157,81),(57,158,82),(58,159,83),(59,160,84),(60,161,85),(61,162,86),(62,163,87)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(63,170),(64,171),(65,172),(66,173),(67,174),(68,175),(69,176),(70,177),(71,178),(72,179),(73,180),(74,181),(75,182),(76,183),(77,184),(78,185),(79,186),(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),(107,152),(108,153),(109,154),(110,155),(111,156),(112,157),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,166),(122,167),(123,168),(124,169)])

186 conjugacy classes

class 1 2A2B2C 3  6 31A···31AD62A···62AD62AE···62CL93A···93AD186A···186AD
order12223631···3162···6262···6293···93186···186
size1133221···11···13···32···22···2

186 irreducible representations

dim1111112222
type+++++
imageC1C2C2C31C62C62S3D6S3×C31S3×C62
kernelS3×C62S3×C31C186D6S3C6C62C31C2C1
# reps121306030113030

Matrix representation of S3×C62 in GL2(𝔽373) generated by

2870
0287
,
0372
1372
,
0372
3720
G:=sub<GL(2,GF(373))| [287,0,0,287],[0,1,372,372],[0,372,372,0] >;

S3×C62 in GAP, Magma, Sage, TeX

S_3\times C_{62}
% in TeX

G:=Group("S3xC62");
// GroupNames label

G:=SmallGroup(372,13);
// by ID

G=gap.SmallGroup(372,13);
# by ID

G:=PCGroup([4,-2,-2,-31,-3,3971]);
// Polycyclic

G:=Group<a,b,c|a^62=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×C62 in TeX

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