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G = S3×C52order 312 = 23·3·13

Direct product of C52 and S3

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

Aliases: S3×C52, D6.C26, C122C26, C1566C2, C26.14D6, Dic32C26, C78.19C22, C31(C2×C52), C399(C2×C4), C2.1(S3×C26), C6.2(C2×C26), (S3×C26).2C2, (Dic3×C13)⋊5C2, SmallGroup(312,33)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C52
C1C3C6C78S3×C26 — S3×C52
C3 — S3×C52
C1C52

Generators and relations for S3×C52
 G = < a,b,c | a52=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C4
3C22
3C26
3C26
3C2×C4
3C52
3C2×C26
3C2×C52

Smallest permutation representation of S3×C52
On 156 points
Generators in S156
(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)
(1 56 134)(2 57 135)(3 58 136)(4 59 137)(5 60 138)(6 61 139)(7 62 140)(8 63 141)(9 64 142)(10 65 143)(11 66 144)(12 67 145)(13 68 146)(14 69 147)(15 70 148)(16 71 149)(17 72 150)(18 73 151)(19 74 152)(20 75 153)(21 76 154)(22 77 155)(23 78 156)(24 79 105)(25 80 106)(26 81 107)(27 82 108)(28 83 109)(29 84 110)(30 85 111)(31 86 112)(32 87 113)(33 88 114)(34 89 115)(35 90 116)(36 91 117)(37 92 118)(38 93 119)(39 94 120)(40 95 121)(41 96 122)(42 97 123)(43 98 124)(44 99 125)(45 100 126)(46 101 127)(47 102 128)(48 103 129)(49 104 130)(50 53 131)(51 54 132)(52 55 133)
(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)

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

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

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

156 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B13A···13L26A···26L26M···26AJ39A···39L52A···52X52Y···52AV78A···78L156A···156X
order1222344446121213···1326···2626···2639···3952···5252···5278···78156···156
size1133211332221···11···13···32···21···13···32···22···2

156 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C4C13C26C26C26C52S3D6C4×S3S3×C13S3×C26S3×C52
kernelS3×C52Dic3×C13C156S3×C26S3×C13C4×S3Dic3C12D6S3C52C26C13C4C2C1
# reps111141212121248112121224

Matrix representation of S3×C52 in GL2(𝔽157) generated by

320
032
,
0156
1156
,
1156
0156
G:=sub<GL(2,GF(157))| [32,0,0,32],[0,1,156,156],[1,0,156,156] >;

S3×C52 in GAP, Magma, Sage, TeX

S_3\times C_{52}
% in TeX

G:=Group("S3xC52");
// GroupNames label

G:=SmallGroup(312,33);
// by ID

G=gap.SmallGroup(312,33);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-3,266,5204]);
// Polycyclic

G:=Group<a,b,c|a^52=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×C52 in TeX

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