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G = S3xC43order 258 = 2·3·43

Direct product of C43 and S3

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

Aliases: S3xC43, C3:C86, C129:3C2, SmallGroup(258,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC43
C1C3C129 — S3xC43
C3 — S3xC43
C1C43

Generators and relations for S3xC43
 G = < a,b,c | a43=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 12 in 8 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, S3, C43, C86, S3xC43
3C2
3C86

Smallest permutation representation of S3xC43
On 129 points
Generators in S129
(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)
(1 112 51)(2 113 52)(3 114 53)(4 115 54)(5 116 55)(6 117 56)(7 118 57)(8 119 58)(9 120 59)(10 121 60)(11 122 61)(12 123 62)(13 124 63)(14 125 64)(15 126 65)(16 127 66)(17 128 67)(18 129 68)(19 87 69)(20 88 70)(21 89 71)(22 90 72)(23 91 73)(24 92 74)(25 93 75)(26 94 76)(27 95 77)(28 96 78)(29 97 79)(30 98 80)(31 99 81)(32 100 82)(33 101 83)(34 102 84)(35 103 85)(36 104 86)(37 105 44)(38 106 45)(39 107 46)(40 108 47)(41 109 48)(42 110 49)(43 111 50)
(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 87)(70 88)(71 89)(72 90)(73 91)(74 92)(75 93)(76 94)(77 95)(78 96)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)

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

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

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

129 conjugacy classes

class 1  2  3 43A···43AP86A···86AP129A···129AP
order12343···4386···86129···129
size1321···13···32···2

129 irreducible representations

dim111122
type+++
imageC1C2C43C86S3S3xC43
kernelS3xC43C129S3C3C43C1
# reps114242142

Matrix representation of S3xC43 in GL2(F1033) generated by

3350
0335
,
10321032
10
,
10
10321032
G:=sub<GL(2,GF(1033))| [335,0,0,335],[1032,1,1032,0],[1,1032,0,1032] >;

S3xC43 in GAP, Magma, Sage, TeX

S_3\times C_{43}
% in TeX

G:=Group("S3xC43");
// GroupNames label

G:=SmallGroup(258,3);
// by ID

G=gap.SmallGroup(258,3);
# by ID

G:=PCGroup([3,-2,-43,-3,1550]);
// Polycyclic

G:=Group<a,b,c|a^43=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3xC43 in TeX

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