direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D39, C39⋊5C6, C39⋊2S3, C32⋊1D13, C3⋊(C3×D13), C13⋊3(C3×S3), (C3×C39)⋊2C2, SmallGroup(234,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C3×D39 |
Generators and relations for C3×D39
G = < a,b,c | a3=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D39
►On 78 points
Generators in S78
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)(40 53 66)(41 54 67)(42 55 68)(43 56 69)(44 57 70)(45 58 71)(46 59 72)(47 60 73)(48 61 74)(49 62 75)(50 63 76)(51 64 77)(52 65 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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)
G:=sub<Sym(78)| (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26)(40,53,66)(41,54,67)(42,55,68)(43,56,69)(44,57,70)(45,58,71)(46,59,72)(47,60,73)(48,61,74)(49,62,75)(50,63,76)(51,64,77)(52,65,78), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)>;
G:=Group( (1,27,14)(2,28,15)(3,29,16)(4,30,17)(5,31,18)(6,32,19)(7,33,20)(8,34,21)(9,35,22)(10,36,23)(11,37,24)(12,38,25)(13,39,26)(40,53,66)(41,54,67)(42,55,68)(43,56,69)(44,57,70)(45,58,71)(46,59,72)(47,60,73)(48,61,74)(49,62,75)(50,63,76)(51,64,77)(52,65,78), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71) );
G=PermutationGroup([[(1,27,14),(2,28,15),(3,29,16),(4,30,17),(5,31,18),(6,32,19),(7,33,20),(8,34,21),(9,35,22),(10,36,23),(11,37,24),(12,38,25),(13,39,26),(40,53,66),(41,54,67),(42,55,68),(43,56,69),(44,57,70),(45,58,71),(46,59,72),(47,60,73),(48,61,74),(49,62,75),(50,63,76),(51,64,77),(52,65,78)], [(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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,48),(24,47),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71)]])
C3×D39 is a maximal subgroup of
C3×S3×D13 D39⋊S3
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 13A | ··· | 13F | 39A | ··· | 39AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 13 | ··· | 13 | 39 | ··· | 39 |
| size | 1 | 39 | 1 | 1 | 2 | 2 | 2 | 39 | 39 | 2 | ··· | 2 | 2 | ··· | 2 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D13 | C3×D13 | D39 | C3×D39 |
| kernel | C3×D39 | C3×C39 | D39 | C39 | C39 | C13 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 6 | 12 | 12 | 24 |
Matrix representation of C3×D39 ►in GL2(𝔽79) generated by
| 55 | 0 |
| 0 | 55 |
| 32 | 0 |
| 0 | 42 |
| 0 | 42 |
| 32 | 0 |
G:=sub<GL(2,GF(79))| [55,0,0,55],[32,0,0,42],[0,32,42,0] >;
C3×D39 in GAP, Magma, Sage, TeX
C_3\times D_{39} % in TeX
G:=Group("C3xD39"); // GroupNames label
G:=SmallGroup(234,13);
// by ID
G=gap.SmallGroup(234,13);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,146,3459]);
// Polycyclic
G:=Group<a,b,c|a^3=b^39=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export