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G = S3×C29order 174 = 2·3·29

Direct product of C29 and S3

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

Aliases: S3×C29, C3⋊C58, C873C2, SmallGroup(174,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C29
C1C3C87 — S3×C29
C3 — S3×C29
C1C29

Generators and relations for S3×C29
 G = < a,b,c | a29=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C58

Smallest permutation representation of S3×C29
On 87 points
Generators in S87
(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)
(1 57 79)(2 58 80)(3 30 81)(4 31 82)(5 32 83)(6 33 84)(7 34 85)(8 35 86)(9 36 87)(10 37 59)(11 38 60)(12 39 61)(13 40 62)(14 41 63)(15 42 64)(16 43 65)(17 44 66)(18 45 67)(19 46 68)(20 47 69)(21 48 70)(22 49 71)(23 50 72)(24 51 73)(25 52 74)(26 53 75)(27 54 76)(28 55 77)(29 56 78)
(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)

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

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), (1,57,79)(2,58,80)(3,30,81)(4,31,82)(5,32,83)(6,33,84)(7,34,85)(8,35,86)(9,36,87)(10,37,59)(11,38,60)(12,39,61)(13,40,62)(14,41,63)(15,42,64)(16,43,65)(17,44,66)(18,45,67)(19,46,68)(20,47,69)(21,48,70)(22,49,71)(23,50,72)(24,51,73)(25,52,74)(26,53,75)(27,54,76)(28,55,77)(29,56,78), (30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80) );

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)], [(1,57,79),(2,58,80),(3,30,81),(4,31,82),(5,32,83),(6,33,84),(7,34,85),(8,35,86),(9,36,87),(10,37,59),(11,38,60),(12,39,61),(13,40,62),(14,41,63),(15,42,64),(16,43,65),(17,44,66),(18,45,67),(19,46,68),(20,47,69),(21,48,70),(22,49,71),(23,50,72),(24,51,73),(25,52,74),(26,53,75),(27,54,76),(28,55,77),(29,56,78)], [(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80)])

87 conjugacy classes

class 1  2  3 29A···29AB58A···58AB87A···87AB
order12329···2958···5887···87
size1321···13···32···2

87 irreducible representations

dim111122
type+++
imageC1C2C29C58S3S3×C29
kernelS3×C29C87S3C3C29C1
# reps112828128

Matrix representation of S3×C29 in GL2(𝔽349) generated by

2630
0263
,
0348
1348
,
01
10
G:=sub<GL(2,GF(349))| [263,0,0,263],[0,1,348,348],[0,1,1,0] >;

S3×C29 in GAP, Magma, Sage, TeX

S_3\times C_{29}
% in TeX

G:=Group("S3xC29");
// GroupNames label

G:=SmallGroup(174,1);
// by ID

G=gap.SmallGroup(174,1);
# by ID

G:=PCGroup([3,-2,-29,-3,1046]);
// Polycyclic

G:=Group<a,b,c|a^29=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×C29 in TeX

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