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G = S3×C37  order 222 = 2·3·37

Direct product of C37 and S3

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

Aliases: S3×C37, C3⋊C74, C111⋊3C2, SmallGroup(222,3)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — S3×C37
Chief series
C1 — C3 — C111 — S3×C37
Lower central
C3 — S3×C37
Upper central
C1 — C37

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

C1
3C2
C3
C37
S3
3C74
C111
S3×C37

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

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

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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