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G = S3×C37order 222 = 2·3·37

Direct product of C37 and S3

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

Aliases: S3×C37, C3⋊C74, C1113C2, SmallGroup(222,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C37
C1C3C111 — S3×C37
C3 — S3×C37
C1C37

Generators and relations for S3×C37
 G = < a,b,c | a37=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C74

Smallest permutation representation of S3×C37
On 111 points
Generators in S111
(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)
(1 109 66)(2 110 67)(3 111 68)(4 75 69)(5 76 70)(6 77 71)(7 78 72)(8 79 73)(9 80 74)(10 81 38)(11 82 39)(12 83 40)(13 84 41)(14 85 42)(15 86 43)(16 87 44)(17 88 45)(18 89 46)(19 90 47)(20 91 48)(21 92 49)(22 93 50)(23 94 51)(24 95 52)(25 96 53)(26 97 54)(27 98 55)(28 99 56)(29 100 57)(30 101 58)(31 102 59)(32 103 60)(33 104 61)(34 105 62)(35 106 63)(36 107 64)(37 108 65)
(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 75)(70 76)(71 77)(72 78)(73 79)(74 80)

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

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

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

111 conjugacy classes

class 1  2  3 37A···37AJ74A···74AJ111A···111AJ
order12337···3774···74111···111
size1321···13···32···2

111 irreducible representations

dim111122
type+++
imageC1C2C37C74S3S3×C37
kernelS3×C37C111S3C3C37C1
# reps113636136

Matrix representation of S3×C37 in GL2(𝔽223) generated by

20
02
,
222222
10
,
10
222222
G:=sub<GL(2,GF(223))| [2,0,0,2],[222,1,222,0],[1,222,0,222] >;

S3×C37 in GAP, Magma, Sage, TeX

S_3\times C_{37}
% in TeX

G:=Group("S3xC37");
// GroupNames label

G:=SmallGroup(222,3);
// by ID

G=gap.SmallGroup(222,3);
# by ID

G:=PCGroup([3,-2,-37,-3,1334]);
// Polycyclic

G:=Group<a,b,c|a^37=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×C37 in TeX

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