Copied to
clipboard

G = S3×C43order 258 = 2·3·43

Direct product of C43 and S3

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

Aliases: S3×C43, C3⋊C86, C1293C2, SmallGroup(258,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C43
C1C3C129 — S3×C43
C3 — S3×C43
C1C43

Generators and relations for S3×C43
 G = < a,b,c | a43=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C86

Smallest permutation representation of S3×C43
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 71)(2 113 72)(3 114 73)(4 115 74)(5 116 75)(6 117 76)(7 118 77)(8 119 78)(9 120 79)(10 121 80)(11 122 81)(12 123 82)(13 124 83)(14 125 84)(15 126 85)(16 127 86)(17 128 44)(18 129 45)(19 87 46)(20 88 47)(21 89 48)(22 90 49)(23 91 50)(24 92 51)(25 93 52)(26 94 53)(27 95 54)(28 96 55)(29 97 56)(30 98 57)(31 99 58)(32 100 59)(33 101 60)(34 102 61)(35 103 62)(36 104 63)(37 105 64)(38 106 65)(39 107 66)(40 108 67)(41 109 68)(42 110 69)(43 111 70)
(44 128)(45 129)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 121)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)

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

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

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

129 conjugacy classes

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

129 irreducible representations

dim111122
type+++
imageC1C2C43C86S3S3×C43
kernelS3×C43C129S3C3C43C1
# reps114242142

Matrix representation of S3×C43 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] >;

S3×C43 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 S3×C43 in TeX

׿
×
𝔽