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G = S3×C31order 186 = 2·3·31

Direct product of C31 and S3

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

Aliases: S3×C31, C3⋊C62, C933C2, SmallGroup(186,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C31
C1C3C93 — S3×C31
C3 — S3×C31
C1C31

Generators and relations for S3×C31
 G = < a,b,c | a31=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C62

Smallest permutation representation of S3×C31
On 93 points
Generators in S93
(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)
(1 64 41)(2 65 42)(3 66 43)(4 67 44)(5 68 45)(6 69 46)(7 70 47)(8 71 48)(9 72 49)(10 73 50)(11 74 51)(12 75 52)(13 76 53)(14 77 54)(15 78 55)(16 79 56)(17 80 57)(18 81 58)(19 82 59)(20 83 60)(21 84 61)(22 85 62)(23 86 32)(24 87 33)(25 88 34)(26 89 35)(27 90 36)(28 91 37)(29 92 38)(30 93 39)(31 63 40)
(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 84)(62 85)

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

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

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

93 conjugacy classes

class 1  2  3 31A···31AD62A···62AD93A···93AD
order12331···3162···6293···93
size1321···13···32···2

93 irreducible representations

dim111122
type+++
imageC1C2C31C62S3S3×C31
kernelS3×C31C93S3C3C31C1
# reps113030130

Matrix representation of S3×C31 in GL2(𝔽373) generated by

2130
0213
,
0372
1372
,
01
10
G:=sub<GL(2,GF(373))| [213,0,0,213],[0,1,372,372],[0,1,1,0] >;

S3×C31 in GAP, Magma, Sage, TeX

S_3\times C_{31}
% in TeX

G:=Group("S3xC31");
// GroupNames label

G:=SmallGroup(186,3);
// by ID

G=gap.SmallGroup(186,3);
# by ID

G:=PCGroup([3,-2,-31,-3,1118]);
// Polycyclic

G:=Group<a,b,c|a^31=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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Subgroup lattice of S3×C31 in TeX

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