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## G = S3×D39order 468 = 22·32·13

### Direct product of S3 and D39

Aliases: S3×D39, C392D6, C31D78, C321D26, C131S32, (S3×C13)⋊S3, (C3×S3)⋊D13, C3⋊D392C2, C31(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 >

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

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 2A 2B 2C 3A 3B 3C 6A 6B 13A ··· 13F 26A ··· 26F 39A ··· 39L 39M ··· 39AD 78A ··· 78L order 1 2 2 2 3 3 3 6 6 13 ··· 13 26 ··· 26 39 ··· 39 39 ··· 39 78 ··· 78 size 1 3 39 117 2 2 4 6 78 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4 6 ··· 6

63 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D6 D13 D26 D39 D78 S32 S3×D13 S3×D39 kernel S3×D39 S3×C39 C3×D39 C3⋊D39 S3×C13 D39 C39 C3×S3 C32 S3 C3 C13 C3 C1 # reps 1 1 1 1 1 1 2 6 6 12 12 1 6 12

Matrix representation of S3×D39 in GL6(𝔽79)

 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 1 0 0 0 0 78 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 34 0 0 0 0 72 18 0 0 0 0 0 0 0 78 0 0 0 0 1 78 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 51 0 0 0 0 65 77 0 0 0 0 0 0 78 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

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