direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C42⋊C3, C42⋊C32, (C4×C12)⋊C3, (C2×C6).2A4, C22.(C3×A4), SmallGroup(144,68)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C3×C42⋊C3 |
Generators and relations for C3×C42⋊C3
G = < a,b,c,d | a3=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Character table of C3×C42⋊C3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4A | 4B | 4C | 4D | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 3 | 1 | 1 | 16 | 16 | 16 | 16 | 16 | 16 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ10 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | ζ65 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×A4 |
| ρ12 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | ζ6 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×A4 |
| ρ13 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | -1 | -1 | 1 | -1+2i | -1-2i | -1-2i | 1 | -1+2i | 1 | 1 | complex lifted from C42⋊C3 |
| ρ14 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | -1 | -1 | -1+2i | 1 | 1 | 1 | -1-2i | 1 | -1+2i | -1-2i | complex lifted from C42⋊C3 |
| ρ15 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | -1 | -1 | -1-2i | 1 | 1 | 1 | -1+2i | 1 | -1-2i | -1+2i | complex lifted from C42⋊C3 |
| ρ16 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | -1 | -1 | 1 | -1-2i | -1+2i | -1+2i | 1 | -1-2i | 1 | 1 | complex lifted from C42⋊C3 |
| ρ17 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | ζ6 | ζ65 | ζ3 | 2ζ43ζ32-ζ32 | 2ζ4ζ3-ζ3 | 2ζ4ζ32-ζ32 | ζ3 | 2ζ43ζ3-ζ3 | ζ32 | ζ32 | complex faithful |
| ρ18 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | ζ6 | ζ65 | ζ3 | 2ζ4ζ32-ζ32 | 2ζ43ζ3-ζ3 | 2ζ43ζ32-ζ32 | ζ3 | 2ζ4ζ3-ζ3 | ζ32 | ζ32 | complex faithful |
| ρ19 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | ζ65 | ζ6 | 2ζ43ζ32-ζ32 | ζ3 | ζ32 | ζ3 | 2ζ4ζ32-ζ32 | ζ32 | 2ζ43ζ3-ζ3 | 2ζ4ζ3-ζ3 | complex faithful |
| ρ20 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | ζ6 | ζ65 | 2ζ4ζ3-ζ3 | ζ32 | ζ3 | ζ32 | 2ζ43ζ3-ζ3 | ζ3 | 2ζ4ζ32-ζ32 | 2ζ43ζ32-ζ32 | complex faithful |
| ρ21 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | ζ65 | ζ6 | 2ζ4ζ32-ζ32 | ζ3 | ζ32 | ζ3 | 2ζ43ζ32-ζ32 | ζ32 | 2ζ4ζ3-ζ3 | 2ζ43ζ3-ζ3 | complex faithful |
| ρ22 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | ζ6 | ζ65 | 2ζ43ζ3-ζ3 | ζ32 | ζ3 | ζ32 | 2ζ4ζ3-ζ3 | ζ3 | 2ζ43ζ32-ζ32 | 2ζ4ζ32-ζ32 | complex faithful |
| ρ23 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | ζ65 | ζ6 | ζ32 | 2ζ4ζ3-ζ3 | 2ζ43ζ32-ζ32 | 2ζ43ζ3-ζ3 | ζ32 | 2ζ4ζ32-ζ32 | ζ3 | ζ3 | complex faithful |
| ρ24 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | ζ65 | ζ6 | ζ32 | 2ζ43ζ3-ζ3 | 2ζ4ζ32-ζ32 | 2ζ4ζ3-ζ3 | ζ32 | 2ζ43ζ32-ζ32 | ζ3 | ζ3 | complex faithful |
►On 36 points
Generators in S36
(1 4 6)(2 10 5)(3 12 8)(7 9 11)(13 21 27)(14 22 28)(15 23 25)(16 24 26)(17 34 31)(18 35 32)(19 36 29)(20 33 30)
(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 3 7 2)(4 12 9 10)(5 6 8 11)(13 15)(14 16)(17 18 19 20)(21 23)(22 24)(25 27)(26 28)(29 30 31 32)(33 34 35 36)
(1 16 29)(2 13 32)(3 15 30)(4 24 19)(5 27 35)(6 26 36)(7 14 31)(8 25 33)(9 22 17)(10 21 18)(11 28 34)(12 23 20)
G:=sub<Sym(36)| (1,4,6)(2,10,5)(3,12,8)(7,9,11)(13,21,27)(14,22,28)(15,23,25)(16,24,26)(17,34,31)(18,35,32)(19,36,29)(20,33,30), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,3,7,2)(4,12,9,10)(5,6,8,11)(13,15)(14,16)(17,18,19,20)(21,23)(22,24)(25,27)(26,28)(29,30,31,32)(33,34,35,36), (1,16,29)(2,13,32)(3,15,30)(4,24,19)(5,27,35)(6,26,36)(7,14,31)(8,25,33)(9,22,17)(10,21,18)(11,28,34)(12,23,20)>;
G:=Group( (1,4,6)(2,10,5)(3,12,8)(7,9,11)(13,21,27)(14,22,28)(15,23,25)(16,24,26)(17,34,31)(18,35,32)(19,36,29)(20,33,30), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,3,7,2)(4,12,9,10)(5,6,8,11)(13,15)(14,16)(17,18,19,20)(21,23)(22,24)(25,27)(26,28)(29,30,31,32)(33,34,35,36), (1,16,29)(2,13,32)(3,15,30)(4,24,19)(5,27,35)(6,26,36)(7,14,31)(8,25,33)(9,22,17)(10,21,18)(11,28,34)(12,23,20) );
G=PermutationGroup([[(1,4,6),(2,10,5),(3,12,8),(7,9,11),(13,21,27),(14,22,28),(15,23,25),(16,24,26),(17,34,31),(18,35,32),(19,36,29),(20,33,30)], [(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,3,7,2),(4,12,9,10),(5,6,8,11),(13,15),(14,16),(17,18,19,20),(21,23),(22,24),(25,27),(26,28),(29,30,31,32),(33,34,35,36)], [(1,16,29),(2,13,32),(3,15,30),(4,24,19),(5,27,35),(6,26,36),(7,14,31),(8,25,33),(9,22,17),(10,21,18),(11,28,34),(12,23,20)]])
C3×C42⋊C3 is a maximal subgroup of
(C4×C12)⋊S3 (C4×C12)⋊C6 C42⋊C3⋊S3 C42⋊3- 1+2 C42⋊He3
C3×C42⋊C3 is a maximal quotient of C42⋊3- 1+2 C122.C3 C42⋊He3
Matrix representation of C3×C42⋊C3 ►in GL3(𝔽13) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 8 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 1 |
| 8 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 8 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(13))| [3,0,0,0,3,0,0,0,3],[8,0,0,0,5,0,0,0,1],[8,0,0,0,12,0,0,0,8],[0,0,1,1,0,0,0,1,0] >;
C3×C42⋊C3 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes C_3
% in TeX
G:=Group("C3xC4^2:C3"); // GroupNames label
G:=SmallGroup(144,68);
// by ID
G=gap.SmallGroup(144,68);
# by ID
G:=PCGroup([6,-3,-3,-2,2,-2,2,326,230,2379,69,2164,3893]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations
Export
Subgroup lattice of C3×C42⋊C3 in TeX
Character table of C3×C42⋊C3 in TeX