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G = S3×D39  order 468 = 22·32·13

Direct product of S3 and D39

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3×D39, C39⋊2D6, C3⋊1D78, C32⋊1D26, C13⋊1S32, (S3×C13)⋊S3, (C3×S3)⋊D13, C3⋊D39⋊2C2, C3⋊1(S3×D13), (S3×C39)⋊1C2, (C3×D39)⋊2C2, (C3×C39)⋊3C22, SmallGroup(468,45)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C39 — S3×D39
Chief series
C1 — C13 — C39 — C3×C39 — C3×D39 — S3×D39
Lower central
C3×C39 — S3×D39
Upper central
C1

Generators and relations for S3×D39
 G = < a,b,c,d | a3=b2=c39=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

C1
3C2
39C2
117C2
C3
C3
2C3
C13
117C22
S3
3C6
13S3
39S3
39C6
39S3
78S3
C32
3C26
3D13
9D13
C39
C39
2C39
39D6
39D6
C3×S3
13C3⋊S3
13C3×S3
9D26
D39
S3×C13
3D39
3C3×D13
3D39
3C78
6D39
C3×C39
13S32
3D78
3S3×D13
C3×D39
C3⋊D39
S3×C39
S3×D39

Smallest permutation representation of S3×D39
►On 78 points
Generators in S78
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)(40 66 53)(41 67 54)(42 68 55)(43 69 56)(44 70 57)(45 71 58)(46 72 59)(47 73 60)(48 74 61)(49 75 62)(50 76 63)(51 77 64)(52 78 65)

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

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

63 conjugacy classes

class 1 2A2B2C3A3B3C6A6B13A···13F26A···26F39A···39L39M···39AD78A···78L
order12223336613···1326···2639···3939···3978···78
size13391172246782···26···62···24···46···6

63 irreducible representations

dim11112222222444
type++++++++++++++
imageC1C2C2C2S3S3D6D13D26D39D78S32S3×D13S3×D39
kernelS3×D39S3×C39C3×D39C3⋊D39S3×C13D39C39C3×S3C32S3C3C13C3C1
# reps11111126612121612

Matrix representation of S3×D39 ►in GL6(𝔽79)

100000
010000
001000
000100
0000781
0000780
,
7800000
0780000
0078000
0007800
000001
000010
,
0340000
72180000
0007800
0017800
000010
000001
,
2510000
65770000
0078100
000100
000010
000001

G:=sub<GL(6,GF(79))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,78,78,0,0,0,0,1,0],[78,0,0,0,0,0,0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,34,18,0,0,0,0,0,0,0,1,0,0,0,0,78,78,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,65,0,0,0,0,51,77,0,0,0,0,0,0,78,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

S3×D39 in GAP, Magma, Sage, TeX 

S_3\times D_{39}
 
% in TeX
 
G:=Group("S3xD39");
 
// GroupNames label
 
G:=SmallGroup(468,45);
 
// by ID
 
G=gap.SmallGroup(468,45);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-13,67,483,10804]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^39=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of S3×D39 in TeX

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