metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C39⋊1C4, C13⋊Dic3, D13.S3, C3⋊(C13⋊C4), (C3×D13).1C2, SmallGroup(156,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C39⋊C4 |
Generators and relations for C39⋊C4
G = < a,b | a39=b4=1, bab-1=a8 >
Character table of C39⋊C4
| class | 1 | 2 | 3 | 4A | 4B | 6 | 13A | 13B | 13C | 39A | 39B | 39C | 39D | 39E | 39F | |
| size | 1 | 13 | 2 | 39 | 39 | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -i | i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 4 | 0 | 4 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | orthogonal lifted from C13⋊C4 |
| ρ8 | 4 | 0 | 4 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | orthogonal lifted from C13⋊C4 |
| ρ9 | 4 | 0 | 4 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | ζ1312+ζ138+ζ135+ζ13 | ζ1312+ζ138+ζ135+ζ13 | ζ1311+ζ1310+ζ133+ζ132 | ζ139+ζ137+ζ136+ζ134 | orthogonal lifted from C13⋊C4 |
| ρ10 | 4 | 0 | -2 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134-ζ139-ζ134 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13-ζ138-ζ135 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132-ζ1311-ζ132 | complex faithful |
| ρ11 | 4 | 0 | -2 | 0 | 0 | 0 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13-ζ138-ζ135 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132-ζ1311-ζ132 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134-ζ139-ζ134 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | complex faithful |
| ρ12 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13-ζ138-ζ135 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13-ζ1312-ζ13 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132-ζ1311-ζ132 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134-ζ139-ζ134 | complex faithful |
| ρ13 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134-ζ139-ζ134 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13-ζ1312-ζ13 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13-ζ138-ζ135 | complex faithful |
| ρ14 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1311+ζ1310+ζ133+ζ132 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13-ζ138-ζ135 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132-ζ1311-ζ132 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134-ζ139-ζ134 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13-ζ1312-ζ13 | complex faithful |
| ρ15 | 4 | 0 | -2 | 0 | 0 | 0 | ζ1312+ζ138+ζ135+ζ13 | ζ139+ζ137+ζ136+ζ134 | ζ1311+ζ1310+ζ133+ζ132 | -ζ32ζ1311+ζ32ζ1310+ζ32ζ133-ζ32ζ132-ζ1311-ζ132 | -ζ32ζ139+ζ32ζ137+ζ32ζ136-ζ32ζ134-ζ139-ζ134 | -ζ3ζ1312+ζ3ζ138+ζ3ζ135-ζ3ζ13-ζ1312-ζ13 | ζ3ζ1312-ζ3ζ138-ζ3ζ135+ζ3ζ13-ζ138-ζ135 | -ζ3ζ1311+ζ3ζ1310+ζ3ζ133-ζ3ζ132-ζ1311-ζ132 | ζ32ζ139-ζ32ζ137-ζ32ζ136+ζ32ζ134-ζ137-ζ136 | complex 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)
(2 6 26 9)(3 11 12 17)(4 16 37 25)(5 21 23 33)(7 31 34 10)(8 36 20 18)(13 22 28 19)(14 27)(15 32 39 35)(24 38 30 29)
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), (2,6,26,9)(3,11,12,17)(4,16,37,25)(5,21,23,33)(7,31,34,10)(8,36,20,18)(13,22,28,19)(14,27)(15,32,39,35)(24,38,30,29)>;
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), (2,6,26,9)(3,11,12,17)(4,16,37,25)(5,21,23,33)(7,31,34,10)(8,36,20,18)(13,22,28,19)(14,27)(15,32,39,35)(24,38,30,29) );
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)], [(2,6,26,9),(3,11,12,17),(4,16,37,25),(5,21,23,33),(7,31,34,10),(8,36,20,18),(13,22,28,19),(14,27),(15,32,39,35),(24,38,30,29)]])
C39⋊C4 is a maximal subgroup of
S3×C13⋊C4 C13⋊Dic9 C3⋊F13 C39⋊Dic3
C39⋊C4 is a maximal quotient of C39⋊C8 C13⋊Dic9 C39⋊Dic3
Matrix representation of C39⋊C4 ►in GL4(𝔽5) generated by
| 2 | 0 | 2 | 4 |
| 2 | 2 | 0 | 4 |
| 3 | 0 | 2 | 0 |
| 0 | 4 | 3 | 3 |
| 1 | 2 | 1 | 0 |
| 0 | 4 | 3 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 4 | 0 |
G:=sub<GL(4,GF(5))| [2,2,3,0,0,2,0,4,2,0,2,3,4,4,0,3],[1,0,0,0,2,4,1,1,1,3,0,4,0,1,0,0] >;
C39⋊C4 in GAP, Magma, Sage, TeX
C_{39}\rtimes C_4 % in TeX
G:=Group("C39:C4"); // GroupNames label
G:=SmallGroup(156,10);
// by ID
G=gap.SmallGroup(156,10);
# by ID
G:=PCGroup([4,-2,-2,-3,-13,8,98,963,1159]);
// Polycyclic
G:=Group<a,b|a^39=b^4=1,b*a*b^-1=a^8>;
// generators/relations
Export
Subgroup lattice of C39⋊C4 in TeX
Character table of C39⋊C4 in TeX