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G = S3×C47order 282 = 2·3·47

Direct product of C47 and S3

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

Aliases: S3×C47, C3⋊C94, C1413C2, SmallGroup(282,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C47
C1C3C141 — S3×C47
C3 — S3×C47
C1C47

Generators and relations for S3×C47
 G = < a,b,c | a47=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C94

Smallest permutation representation of S3×C47
On 141 points
Generators in S141
(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)
(1 79 125)(2 80 126)(3 81 127)(4 82 128)(5 83 129)(6 84 130)(7 85 131)(8 86 132)(9 87 133)(10 88 134)(11 89 135)(12 90 136)(13 91 137)(14 92 138)(15 93 139)(16 94 140)(17 48 141)(18 49 95)(19 50 96)(20 51 97)(21 52 98)(22 53 99)(23 54 100)(24 55 101)(25 56 102)(26 57 103)(27 58 104)(28 59 105)(29 60 106)(30 61 107)(31 62 108)(32 63 109)(33 64 110)(34 65 111)(35 66 112)(36 67 113)(37 68 114)(38 69 115)(39 70 116)(40 71 117)(41 72 118)(42 73 119)(43 74 120)(44 75 121)(45 76 122)(46 77 123)(47 78 124)
(48 141)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 101)(56 102)(57 103)(58 104)(59 105)(60 106)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)

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

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

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

141 conjugacy classes

class 1  2  3 47A···47AT94A···94AT141A···141AT
order12347···4794···94141···141
size1321···13···32···2

141 irreducible representations

dim111122
type+++
imageC1C2C47C94S3S3×C47
kernelS3×C47C141S3C3C47C1
# reps114646146

Matrix representation of S3×C47 in GL2(𝔽283) generated by

1810
0181
,
282282
10
,
01
10
G:=sub<GL(2,GF(283))| [181,0,0,181],[282,1,282,0],[0,1,1,0] >;

S3×C47 in GAP, Magma, Sage, TeX

S_3\times C_{47}
% in TeX

G:=Group("S3xC47");
// GroupNames label

G:=SmallGroup(282,1);
// by ID

G=gap.SmallGroup(282,1);
# by ID

G:=PCGroup([3,-2,-47,-3,1694]);
// Polycyclic

G:=Group<a,b,c|a^47=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×C47 in TeX

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