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G = S3×C23order 138 = 2·3·23

Direct product of C23 and S3

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

Aliases: S3×C23, C3⋊C46, C693C2, SmallGroup(138,1)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C23
C1C3C69 — S3×C23
C3 — S3×C23
C1C23

Generators and relations for S3×C23
 G = < a,b,c | a23=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C46

Smallest permutation representation of S3×C23
On 69 points
Generators in S69
(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)
(1 34 66)(2 35 67)(3 36 68)(4 37 69)(5 38 47)(6 39 48)(7 40 49)(8 41 50)(9 42 51)(10 43 52)(11 44 53)(12 45 54)(13 46 55)(14 24 56)(15 25 57)(16 26 58)(17 27 59)(18 28 60)(19 29 61)(20 30 62)(21 31 63)(22 32 64)(23 33 65)
(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 47)(39 48)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)

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

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

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

69 conjugacy classes

class 1  2  3 23A···23V46A···46V69A···69V
order12323···2346···4669···69
size1321···13···32···2

69 irreducible representations

dim111122
type+++
imageC1C2C23C46S3S3×C23
kernelS3×C23C69S3C3C23C1
# reps112222122

Matrix representation of S3×C23 in GL2(𝔽47) generated by

30
03
,
043
1246
,
143
046
G:=sub<GL(2,GF(47))| [3,0,0,3],[0,12,43,46],[1,0,43,46] >;

S3×C23 in GAP, Magma, Sage, TeX

S_3\times C_{23}
% in TeX

G:=Group("S3xC23");
// GroupNames label

G:=SmallGroup(138,1);
// by ID

G=gap.SmallGroup(138,1);
# by ID

G:=PCGroup([3,-2,-23,-3,830]);
// Polycyclic

G:=Group<a,b,c|a^23=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×C23 in TeX

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