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G = S3×C26order 156 = 22·3·13

Direct product of C26 and S3

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

Aliases: S3×C26, C6⋊C26, C783C2, C394C22, C3⋊(C2×C26), SmallGroup(156,16)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C26
C1C3C39S3×C13 — S3×C26
C3 — S3×C26
C1C26

Generators and relations for S3×C26
 G = < a,b,c | a26=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C26
3C26
3C2×C26

Smallest permutation representation of S3×C26
On 78 points
Generators in S78
(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)
(1 50 53)(2 51 54)(3 52 55)(4 27 56)(5 28 57)(6 29 58)(7 30 59)(8 31 60)(9 32 61)(10 33 62)(11 34 63)(12 35 64)(13 36 65)(14 37 66)(15 38 67)(16 39 68)(17 40 69)(18 41 70)(19 42 71)(20 43 72)(21 44 73)(22 45 74)(23 46 75)(24 47 76)(25 48 77)(26 49 78)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)

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

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

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

S3×C26 is a maximal subgroup of   C39⋊D4  C13⋊D12

78 conjugacy classes

class 1 2A2B2C 3  6 13A···13L26A···26L26M···26AJ39A···39L78A···78L
order12223613···1326···2626···2639···3978···78
size1133221···11···13···32···22···2

78 irreducible representations

dim1111112222
type+++++
imageC1C2C2C13C26C26S3D6S3×C13S3×C26
kernelS3×C26S3×C13C78D6S3C6C26C13C2C1
# reps121122412111212

Matrix representation of S3×C26 in GL4(𝔽79) generated by

38000
07800
0010
0001
,
1000
0100
007878
0010
,
78000
0100
0010
007878
G:=sub<GL(4,GF(79))| [38,0,0,0,0,78,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,78,1,0,0,78,0],[78,0,0,0,0,1,0,0,0,0,1,78,0,0,0,78] >;

S3×C26 in GAP, Magma, Sage, TeX

S_3\times C_{26}
% in TeX

G:=Group("S3xC26");
// GroupNames label

G:=SmallGroup(156,16);
// by ID

G=gap.SmallGroup(156,16);
# by ID

G:=PCGroup([4,-2,-2,-13,-3,1667]);
// Polycyclic

G:=Group<a,b,c|a^26=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×C26 in TeX

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