direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3xD39, C39:2D6, C3:1D78, C32:1D26, C13:1S32, (S3xC13):S3, (C3xS3):D13, C3:D39:2C2, C3:1(S3xD13), (S3xC39):1C2, (C3xD39):2C2, (C3xC39):3C22, SmallGroup(468,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C3xC39 — S3xD39 |
Generators and relations for S3xD39
G = < a,b,c,d | a3=b2=c39=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Quotients: C1, C2, C22, S3, D6, D13, S32, D26, D39, S3xD13, D78, S3xD39
3C2
39C2
117C2
2C3
117C22
3C6
13S3
39S3
39C6
39S3
78S3
3C26
3D13
9D13
2C39
39D6
39D6
13C3:S3
13C3xS3
9D26
3D39
3D39
3C78
6D39
13S32
3D78
Smallest permutation representation of S3xD39
►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 | S3xD13 | S3xD39 |
| kernel | S3xD39 | S3xC39 | C3xD39 | C3:D39 | S3xC13 | D39 | C39 | C3xS3 | C32 | S3 | C3 | C13 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 12 | 1 | 6 | 12 |
Matrix representation of S3xD39 ►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] >;
S3xD39 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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