direct product, non-abelian, soluble, monomial
Aliases: C3×A4⋊C4, A4⋊C12, C6.10S4, (C2×A4).C6, (C3×A4)⋊1C4, C2.1(C3×S4), C23.(C3×S3), (C6×A4).1C2, (C2×C6)⋊1Dic3, C22⋊(C3×Dic3), (C22×C6).1S3, SmallGroup(144,123)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C3×A4⋊C4 |
Generators and relations for C3×A4⋊C4
G = < a,b,c,d,e | a3=b2=c2=d3=e4=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=d-1 >
Character table of C3×A4⋊C4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | 8 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 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 | -1 | -1 | i | -i | i | i | -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 | -1 | -1 | -i | i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | -i | i | i | -i | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | -1 | ζ65 | ζ6 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | i | -i | -i | i | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | -1 | ζ6 | ζ65 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | i | -i | -i | i | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ32 | -1 | ζ65 | ζ6 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | -i | i | i | -i | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ3 | -1 | ζ6 | ζ65 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | linear of order 12 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | -2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 1+√-3 | 1-√-3 | -1+√-3 | -1-√-3 | 1 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Dic3 |
| ρ16 | 2 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ17 | 2 | -2 | -2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 1-√-3 | 1+√-3 | -1-√-3 | -1+√-3 | 1 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Dic3 |
| ρ18 | 2 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | -1+√-3 | -1+√-3 | -1-√-3 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ19 | 3 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from S4 |
| ρ21 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | ζ6 | ζ65 | ζ3 | ζ65 | ζ32 | ζ6 | ζ32 | ζ3 | complex lifted from C3×S4 |
| ρ22 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | ζ65 | ζ6 | ζ32 | ζ6 | ζ3 | ζ65 | ζ3 | ζ32 | complex lifted from C3×S4 |
| ρ23 | 3 | 3 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | ζ32 | ζ3 | ζ65 | ζ3 | ζ6 | ζ32 | ζ6 | ζ65 | complex lifted from C3×S4 |
| ρ24 | 3 | 3 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | ζ3 | ζ32 | ζ6 | ζ32 | ζ65 | ζ3 | ζ65 | ζ6 | complex lifted from C3×S4 |
| ρ25 | 3 | -3 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | -i | -i | i | i | -3 | -3 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | -i | i | i | -i | -i | i | i | -i | complex lifted from A4⋊C4 |
| ρ26 | 3 | -3 | 1 | -1 | 3 | 3 | 0 | 0 | 0 | i | i | -i | -i | -3 | -3 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | i | -i | -i | i | i | -i | -i | i | complex lifted from A4⋊C4 |
| ρ27 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -i | -i | i | i | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | ζ6 | ζ65 | 0 | 0 | 0 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | complex faithful |
| ρ28 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -i | -i | i | i | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | ζ65 | ζ6 | 0 | 0 | 0 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | complex faithful |
| ρ29 | 3 | -3 | 1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | i | i | -i | -i | 3+3√-3/2 | 3-3√-3/2 | ζ3 | ζ32 | ζ6 | ζ65 | 0 | 0 | 0 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | complex faithful |
| ρ30 | 3 | -3 | 1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | i | i | -i | -i | 3-3√-3/2 | 3+3√-3/2 | ζ32 | ζ3 | ζ65 | ζ6 | 0 | 0 | 0 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | complex faithful |
►On 36 points
Generators in S36
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)
(1 4)(2 3)(5 11)(6 8)(7 9)(10 12)(13 14)(15 16)(17 23)(18 20)(19 21)(22 24)(25 26)(27 28)(29 35)(30 32)(31 33)(34 36)
(1 3)(2 4)(5 9)(6 10)(7 11)(8 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)(25 27)(26 28)(29 33)(30 34)(31 35)(32 36)
(1 5 10)(2 11 6)(3 7 12)(4 9 8)(13 21 20)(14 17 22)(15 23 18)(16 19 24)(25 33 32)(26 29 34)(27 35 30)(28 31 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)
G:=sub<Sym(36)| (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24)(25,26)(27,28)(29,35)(30,32)(31,33)(34,36), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24)(25,33,32)(26,29,34)(27,35,30)(28,31,36), (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)>;
G:=Group( (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24), (1,4)(2,3)(5,11)(6,8)(7,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,21)(22,24)(25,26)(27,28)(29,35)(30,32)(31,33)(34,36), (1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36), (1,5,10)(2,11,6)(3,7,12)(4,9,8)(13,21,20)(14,17,22)(15,23,18)(16,19,24)(25,33,32)(26,29,34)(27,35,30)(28,31,36), (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) );
G=PermutationGroup([[(1,26,14),(2,27,15),(3,28,16),(4,25,13),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24)], [(1,4),(2,3),(5,11),(6,8),(7,9),(10,12),(13,14),(15,16),(17,23),(18,20),(19,21),(22,24),(25,26),(27,28),(29,35),(30,32),(31,33),(34,36)], [(1,3),(2,4),(5,9),(6,10),(7,11),(8,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24),(25,27),(26,28),(29,33),(30,34),(31,35),(32,36)], [(1,5,10),(2,11,6),(3,7,12),(4,9,8),(13,21,20),(14,17,22),(15,23,18),(16,19,24),(25,33,32),(26,29,34),(27,35,30),(28,31,36)], [(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)]])
C3×A4⋊C4 is a maximal subgroup of
Dic3.S4 Dic3⋊2S4 A4⋊D12 C12×S4 C62⋊6Dic3
C3×A4⋊C4 is a maximal quotient of C62.Dic3 C62⋊5Dic3
Matrix representation of C3×A4⋊C4 ►in GL3(𝔽13) generated by
| 9 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 0 | 2 | 0 |
| 0 | 0 | 1 |
| 7 | 0 | 0 |
| 5 | 0 | 0 |
| 0 | 0 | 5 |
| 0 | 5 | 0 |
G:=sub<GL(3,GF(13))| [9,0,0,0,9,0,0,0,9],[12,0,0,0,1,0,0,0,12],[1,0,0,0,12,0,0,0,12],[0,0,7,2,0,0,0,1,0],[5,0,0,0,0,5,0,5,0] >;
C3×A4⋊C4 in GAP, Magma, Sage, TeX
C_3\times A_4\rtimes C_4
% in TeX
G:=Group("C3xA4:C4"); // GroupNames label
G:=SmallGroup(144,123);
// by ID
G=gap.SmallGroup(144,123);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-2,2,36,579,2164,202,1301,347]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^3=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of C3×A4⋊C4 in TeX
Character table of C3×A4⋊C4 in TeX