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G = S3×C41order 246 = 2·3·41

Direct product of C41 and S3

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

Aliases: S3×C41, C3⋊C82, C1233C2, SmallGroup(246,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C41
C1C3C123 — S3×C41
C3 — S3×C41
C1C41

Generators and relations for S3×C41
 G = < a,b,c | a41=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C82

Smallest permutation representation of S3×C41
On 123 points
Generators in S123
(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)
(1 120 66)(2 121 67)(3 122 68)(4 123 69)(5 83 70)(6 84 71)(7 85 72)(8 86 73)(9 87 74)(10 88 75)(11 89 76)(12 90 77)(13 91 78)(14 92 79)(15 93 80)(16 94 81)(17 95 82)(18 96 42)(19 97 43)(20 98 44)(21 99 45)(22 100 46)(23 101 47)(24 102 48)(25 103 49)(26 104 50)(27 105 51)(28 106 52)(29 107 53)(30 108 54)(31 109 55)(32 110 56)(33 111 57)(34 112 58)(35 113 59)(36 114 60)(37 115 61)(38 116 62)(39 117 63)(40 118 64)(41 119 65)
(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 121)(68 122)(69 123)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)

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

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

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

123 conjugacy classes

class 1  2  3 41A···41AN82A···82AN123A···123AN
order12341···4182···82123···123
size1321···13···32···2

123 irreducible representations

dim111122
type+++
imageC1C2C41C82S3S3×C41
kernelS3×C41C123S3C3C41C1
# reps114040140

Matrix representation of S3×C41 in GL3(𝔽739) generated by

63100
010
001
,
100
0738738
010
,
73800
010
0738738
G:=sub<GL(3,GF(739))| [631,0,0,0,1,0,0,0,1],[1,0,0,0,738,1,0,738,0],[738,0,0,0,1,738,0,0,738] >;

S3×C41 in GAP, Magma, Sage, TeX

S_3\times C_{41}
% in TeX

G:=Group("S3xC41");
// GroupNames label

G:=SmallGroup(246,1);
// by ID

G=gap.SmallGroup(246,1);
# by ID

G:=PCGroup([3,-2,-41,-3,1478]);
// Polycyclic

G:=Group<a,b,c|a^41=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×C41 in TeX

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