metacyclic, supersoluble, monomial, A-group
Aliases: D39⋊C3, C39⋊1C6, C13⋊C3⋊S3, C13⋊(C3×S3), C3⋊(C13⋊C6), (C3×C13⋊C3)⋊1C2, SmallGroup(234,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — D39⋊C3 |
Generators and relations for D39⋊C3
G = < a,b,c | a39=b2=c3=1, bab=a-1, cac-1=a22, cbc-1=a21b >
Character table of D39⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 13A | 13B | 39A | 39B | 39C | 39D | |
| size | 1 | 39 | 2 | 13 | 13 | 26 | 26 | 39 | 39 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ11 | 6 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ12 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 | -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 | -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 | orthogonal faithful |
| ρ13 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 | -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 | ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 | orthogonal faithful |
| ρ14 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 | ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 | -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 | -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 | orthogonal faithful |
| ρ15 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -ζ32ζ1312-ζ32ζ1310+ζ32ζ139-ζ32ζ134+ζ32ζ133+ζ32ζ13-ζ1312-ζ1310-ζ134 | ζ3ζ1311+ζ3ζ138+ζ3ζ137-ζ3ζ136-ζ3ζ135-ζ3ζ132-ζ136-ζ135-ζ132 | ζ32ζ1312+ζ32ζ1310-ζ32ζ139+ζ32ζ134-ζ32ζ133-ζ32ζ13-ζ139-ζ133-ζ13 | -ζ3ζ1311-ζ3ζ138-ζ3ζ137+ζ3ζ136+ζ3ζ135+ζ3ζ132-ζ1311-ζ138-ζ137 | orthogonal faithful |
►On 39 points
Generators in S39
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)
(2 17 23)(3 33 6)(4 10 28)(5 26 11)(7 19 16)(8 35 38)(9 12 21)(13 37 31)(15 30 36)(18 39 24)(20 32 29)(22 25 34)
G:=sub<Sym(39)| (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21), (2,17,23)(3,33,6)(4,10,28)(5,26,11)(7,19,16)(8,35,38)(9,12,21)(13,37,31)(15,30,36)(18,39,24)(20,32,29)(22,25,34)>;
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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21), (2,17,23)(3,33,6)(4,10,28)(5,26,11)(7,19,16)(8,35,38)(9,12,21)(13,37,31)(15,30,36)(18,39,24)(20,32,29)(22,25,34) );
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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21)], [(2,17,23),(3,33,6),(4,10,28),(5,26,11),(7,19,16),(8,35,38),(9,12,21),(13,37,31),(15,30,36),(18,39,24),(20,32,29),(22,25,34)]])
D39⋊C3 is a maximal subgroup of
S3×C13⋊C6
D39⋊C3 is a maximal quotient of C39⋊3C12
Matrix representation of D39⋊C3 ►in GL6(𝔽79)
| 76 | 31 | 61 | 12 | 55 | 8 |
| 48 | 6 | 64 | 39 | 56 | 14 |
| 25 | 31 | 49 | 9 | 70 | 72 |
| 51 | 3 | 3 | 15 | 77 | 21 |
| 9 | 34 | 46 | 27 | 46 | 14 |
| 48 | 18 | 67 | 24 | 71 | 5 |
| 10 | 3 | 19 | 48 | 3 | 76 |
| 11 | 53 | 0 | 73 | 31 | 48 |
| 73 | 6 | 32 | 48 | 54 | 76 |
| 4 | 63 | 60 | 76 | 28 | 63 |
| 4 | 66 | 35 | 45 | 69 | 76 |
| 74 | 69 | 76 | 60 | 31 | 76 |
| 0 | 0 | 64 | 13 | 1 | 0 |
| 1 | 0 | 66 | 29 | 0 | 0 |
| 0 | 0 | 64 | 77 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 1 |
| 0 | 1 | 65 | 12 | 0 | 0 |
| 0 | 0 | 65 | 14 | 0 | 0 |
G:=sub<GL(6,GF(79))| [76,48,25,51,9,48,31,6,31,3,34,18,61,64,49,3,46,67,12,39,9,15,27,24,55,56,70,77,46,71,8,14,72,21,14,5],[10,11,73,4,4,74,3,53,6,63,66,69,19,0,32,60,35,76,48,73,48,76,45,60,3,31,54,28,69,31,76,48,76,63,76,76],[0,1,0,0,0,0,0,0,0,0,1,0,64,66,64,1,65,65,13,29,77,15,12,14,1,0,0,0,0,0,0,0,0,1,0,0] >;
D39⋊C3 in GAP, Magma, Sage, TeX
D_{39}\rtimes C_3 % in TeX
G:=Group("D39:C3"); // GroupNames label
G:=SmallGroup(234,9);
// by ID
G=gap.SmallGroup(234,9);
# by ID
G:=PCGroup([4,-2,-3,-3,-13,146,3459,439]);
// Polycyclic
G:=Group<a,b,c|a^39=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^22,c*b*c^-1=a^21*b>;
// generators/relations
Export
Subgroup lattice of D39⋊C3 in TeX
Character table of D39⋊C3 in TeX