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G = S3×C61order 366 = 2·3·61

Direct product of C61 and S3

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

Aliases: S3×C61, C3⋊C122, C1833C2, SmallGroup(366,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C61
C1C3C183 — S3×C61
C3 — S3×C61
C1C61

Generators and relations for S3×C61
 G = < a,b,c | a61=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C122

Smallest permutation representation of S3×C61
On 183 points
Generators in S183
(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)
(1 159 115)(2 160 116)(3 161 117)(4 162 118)(5 163 119)(6 164 120)(7 165 121)(8 166 122)(9 167 62)(10 168 63)(11 169 64)(12 170 65)(13 171 66)(14 172 67)(15 173 68)(16 174 69)(17 175 70)(18 176 71)(19 177 72)(20 178 73)(21 179 74)(22 180 75)(23 181 76)(24 182 77)(25 183 78)(26 123 79)(27 124 80)(28 125 81)(29 126 82)(30 127 83)(31 128 84)(32 129 85)(33 130 86)(34 131 87)(35 132 88)(36 133 89)(37 134 90)(38 135 91)(39 136 92)(40 137 93)(41 138 94)(42 139 95)(43 140 96)(44 141 97)(45 142 98)(46 143 99)(47 144 100)(48 145 101)(49 146 102)(50 147 103)(51 148 104)(52 149 105)(53 150 106)(54 151 107)(55 152 108)(56 153 109)(57 154 110)(58 155 111)(59 156 112)(60 157 113)(61 158 114)
(62 167)(63 168)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 183)(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)(119 163)(120 164)(121 165)(122 166)

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

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

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

183 conjugacy classes

class 1  2  3 61A···61BH122A···122BH183A···183BH
order12361···61122···122183···183
size1321···13···32···2

183 irreducible representations

dim111122
type+++
imageC1C2C61C122S3S3×C61
kernelS3×C61C183S3C3C61C1
# reps116060160

Matrix representation of S3×C61 in GL2(𝔽367) generated by

2040
0204
,
366366
10
,
10
366366
G:=sub<GL(2,GF(367))| [204,0,0,204],[366,1,366,0],[1,366,0,366] >;

S3×C61 in GAP, Magma, Sage, TeX

S_3\times C_{61}
% in TeX

G:=Group("S3xC61");
// GroupNames label

G:=SmallGroup(366,3);
// by ID

G=gap.SmallGroup(366,3);
# by ID

G:=PCGroup([3,-2,-61,-3,2198]);
// Polycyclic

G:=Group<a,b,c|a^61=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×C61 in TeX

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