direct product, metabelian, soluble, monomial, A-group
Aliases: Dic3xA4, C3:(C4xA4), (C2xC6):C12, (C3xA4):3C4, C6.3(C2xA4), C2.1(S3xA4), (C22xC6).C6, (C2xA4).2S3, (C6xA4).3C2, C23.2(C3xS3), (C22xDic3):C3, C22:2(C3xDic3), SmallGroup(144,129)
Series: Derived ►Chief ►Lower central ►Upper central
| C2xC6 — Dic3xA4 |
Generators and relations for Dic3xA4
G = < a,b,c,d,e | a6=c2=d2=e3=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, A4, C3xS3, C2xA4, C3xDic3, C4xA4, S3xA4, Dic3xA4
Character table of Dic3xA4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 2 | 4 | 4 | 8 | 8 | 3 | 3 | 9 | 9 | 2 | 4 | 4 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | 1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | i | -i | i | -i | -1 | ζ6 | ζ65 | -1 | 1 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | linear of order 12 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | -i | i | -i | i | -1 | ζ6 | ζ65 | -1 | 1 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | i | -i | i | -i | -1 | ζ65 | ζ6 | -1 | 1 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | -i | i | -i | i | -1 | ζ65 | ζ6 | -1 | 1 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | 2 | 2 | 2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 1 | 1-√-3 | 1+√-3 | 1 | -1 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | complex lifted from C3xDic3 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 1 | 1+√-3 | 1-√-3 | 1 | -1 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | complex lifted from C3xDic3 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ19 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ20 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ21 | 3 | -3 | -1 | 1 | 3 | 0 | 0 | 0 | 0 | 3i | -3i | -i | i | -3 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4xA4 |
| ρ22 | 3 | -3 | -1 | 1 | 3 | 0 | 0 | 0 | 0 | -3i | 3i | i | -i | -3 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4xA4 |
| ρ23 | 6 | 6 | -2 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xA4 |
| ρ24 | 6 | -6 | -2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 36 points
Generators in S36
(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)
(1 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 17 11)(2 18 12)(3 13 7)(4 14 8)(5 15 9)(6 16 10)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
G:=sub<Sym(36)| (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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,17,11)(2,18,12)(3,13,7)(4,14,8)(5,15,9)(6,16,10)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)>;
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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,17,11)(2,18,12)(3,13,7)(4,14,8)(5,15,9)(6,16,10)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30) );
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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36)], [(1,17,11),(2,18,12),(3,13,7),(4,14,8),(5,15,9),(6,16,10),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)]])
Dic3xA4 is a maximal subgroup of
Dic3.S4 Dic3:2S4 Dic3:S4 C4xS3xA4 Dic9:A4 C62:4C12
Dic3xA4 is a maximal quotient of SL2(F3).Dic3 Dic9:A4 C62:4C12
Matrix representation of Dic3xA4 ►in GL5(F13)
| 1 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 5 | 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [1,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,0,0,0,0,8,8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
Dic3xA4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times A_4 % in TeX
G:=Group("Dic3xA4"); // GroupNames label
G:=SmallGroup(144,129);
// by ID
G=gap.SmallGroup(144,129);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,2,-3,36,441,190,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=c^2=d^2=e^3=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of Dic3xA4 in TeX
Character table of Dic3xA4 in TeX