direct product, metabelian, soluble, monomial
Aliases: C3×C42⋊C6, (C4×C12)⋊2C6, C42⋊C3⋊1C6, C42⋊1(C3×C6), C42⋊2C2⋊C32, C23.1(C3×A4), C22.3(C6×A4), (C22×C6).5A4, (C3×C42⋊C3)⋊2C2, (C3×C42⋊2C2)⋊C3, (C2×C6).11(C2×A4), SmallGroup(288,635)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C3×C42⋊C6 |
Generators and relations for C3×C42⋊C6
G = < a,b,c,d | a3=b4=c4=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=c-1, dcd-1=b-1c >
Character table of C3×C42⋊C6
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 3 | 4 | 1 | 1 | 16 | 16 | 16 | 16 | 16 | 16 | 6 | 6 | 12 | 3 | 3 | 4 | 4 | 16 | 16 | 16 | 16 | 16 | 16 | 6 | 6 | 6 | 6 | 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 | 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 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | -1 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | -1 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ10 | 1 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ12 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ13 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ14 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ15 | 1 | 1 | -1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ16 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ17 | 1 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 6 |
| ρ18 | 1 | 1 | -1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ19 | 3 | 3 | -3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 3 | 3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from C2×A4 |
| ρ20 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ21 | 3 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ32 | complex lifted from C6×A4 |
| ρ22 | 3 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ3 | complex lifted from C6×A4 |
| ρ23 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | complex lifted from C3×A4 |
| ρ24 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | complex lifted from C3×A4 |
| ρ25 | 6 | -2 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | complex lifted from C42⋊C6 |
| ρ26 | 6 | -2 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | complex lifted from C42⋊C6 |
| ρ27 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ4ζ3 | 2ζ43ζ32 | 2ζ4ζ32 | 0 | 0 | complex faithful |
| ρ28 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ43ζ32 | 2ζ4ζ3 | 2ζ43ζ3 | 0 | 0 | complex faithful |
| ρ29 | 6 | -2 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ43ζ3 | 2ζ4ζ32 | 2ζ43ζ32 | 0 | 0 | complex faithful |
| ρ30 | 6 | -2 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ4ζ32 | 2ζ43ζ3 | 2ζ4ζ3 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 2 3)(4 8 11)(5 9 12)(6 7 10)(13 19 29)(14 20 30)(15 21 25)(16 22 26)(17 23 27)(18 24 28)(31 37 47)(32 38 48)(33 39 43)(34 40 44)(35 41 45)(36 42 46)
(1 25 7 38)(2 15 10 48)(3 21 6 32)(4 41 12 28)(5 18 8 45)(9 24 11 35)(13 40 16 30)(14 19 44 22)(17 36 47 33)(20 29 34 26)(23 42 31 39)(27 46 37 43)
(1 47 12 14)(2 31 5 20)(3 37 9 30)(4 44 7 17)(6 27 11 40)(8 34 10 23)(13 21 43 24)(15 39 18 29)(16 32 46 35)(19 25 33 28)(22 38 36 41)(26 48 42 45)
(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 40 41 42)(43 44 45 46 47 48)
G:=sub<Sym(48)| (1,2,3)(4,8,11)(5,9,12)(6,7,10)(13,19,29)(14,20,30)(15,21,25)(16,22,26)(17,23,27)(18,24,28)(31,37,47)(32,38,48)(33,39,43)(34,40,44)(35,41,45)(36,42,46), (1,25,7,38)(2,15,10,48)(3,21,6,32)(4,41,12,28)(5,18,8,45)(9,24,11,35)(13,40,16,30)(14,19,44,22)(17,36,47,33)(20,29,34,26)(23,42,31,39)(27,46,37,43), (1,47,12,14)(2,31,5,20)(3,37,9,30)(4,44,7,17)(6,27,11,40)(8,34,10,23)(13,21,43,24)(15,39,18,29)(16,32,46,35)(19,25,33,28)(22,38,36,41)(26,48,42,45), (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)(37,38,39,40,41,42)(43,44,45,46,47,48)>;
G:=Group( (1,2,3)(4,8,11)(5,9,12)(6,7,10)(13,19,29)(14,20,30)(15,21,25)(16,22,26)(17,23,27)(18,24,28)(31,37,47)(32,38,48)(33,39,43)(34,40,44)(35,41,45)(36,42,46), (1,25,7,38)(2,15,10,48)(3,21,6,32)(4,41,12,28)(5,18,8,45)(9,24,11,35)(13,40,16,30)(14,19,44,22)(17,36,47,33)(20,29,34,26)(23,42,31,39)(27,46,37,43), (1,47,12,14)(2,31,5,20)(3,37,9,30)(4,44,7,17)(6,27,11,40)(8,34,10,23)(13,21,43,24)(15,39,18,29)(16,32,46,35)(19,25,33,28)(22,38,36,41)(26,48,42,45), (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)(37,38,39,40,41,42)(43,44,45,46,47,48) );
G=PermutationGroup([[(1,2,3),(4,8,11),(5,9,12),(6,7,10),(13,19,29),(14,20,30),(15,21,25),(16,22,26),(17,23,27),(18,24,28),(31,37,47),(32,38,48),(33,39,43),(34,40,44),(35,41,45),(36,42,46)], [(1,25,7,38),(2,15,10,48),(3,21,6,32),(4,41,12,28),(5,18,8,45),(9,24,11,35),(13,40,16,30),(14,19,44,22),(17,36,47,33),(20,29,34,26),(23,42,31,39),(27,46,37,43)], [(1,47,12,14),(2,31,5,20),(3,37,9,30),(4,44,7,17),(6,27,11,40),(8,34,10,23),(13,21,43,24),(15,39,18,29),(16,32,46,35),(19,25,33,28),(22,38,36,41),(26,48,42,45)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48)]])
Matrix representation of C3×C42⋊C6 ►in GL9(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 5 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 5 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 8 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 12 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 8 | 11 | 11 | 10 | 5 |
| 8 | 9 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 8 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 3 | 3 | 2 | 5 |
| 0 | 0 | 0 | 11 | 8 | 11 | 11 | 10 | 5 |
| 0 | 0 | 0 | 2 | 0 | 10 | 2 | 3 | 8 |
| 0 | 0 | 0 | 5 | 0 | 5 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 8 | 8 | 0 | 0 | 12 | 0 |
| 7 | 7 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 4 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 12 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 10 | 2 | 3 | 8 |
| 0 | 0 | 0 | 2 | 5 | 2 | 10 | 11 | 8 |
| 0 | 0 | 0 | 5 | 5 | 0 | 12 | 0 | 0 |
G:=sub<GL(9,GF(13))| [9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[12,4,5,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,5,0,8,12,0,11,0,0,0,5,0,8,12,1,8,0,0,0,0,0,0,12,0,11,0,0,0,12,1,0,0,0,11,0,0,0,0,0,12,0,0,10,0,0,0,0,0,0,0,0,5],[8,9,7,0,0,0,0,0,0,9,8,7,0,0,0,0,0,0,11,11,9,0,0,0,0,0,0,0,0,0,11,11,2,5,0,8,0,0,0,0,8,0,0,0,8,0,0,0,3,11,10,5,0,0,0,0,0,3,11,2,0,0,0,0,0,0,2,10,3,0,0,12,0,0,0,5,5,8,12,1,0],[7,4,5,0,0,0,0,0,0,7,0,4,0,0,0,0,0,0,8,0,6,0,0,0,0,0,0,0,0,0,1,12,0,2,2,5,0,0,0,0,12,1,0,5,5,0,0,0,0,12,0,10,2,0,0,0,0,0,0,0,2,10,12,0,0,0,0,0,0,3,11,0,0,0,0,0,0,0,8,8,0] >;
C3×C42⋊C6 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes C_6
% in TeX
G:=Group("C3xC4^2:C6"); // GroupNames label
G:=SmallGroup(288,635);
// by ID
G=gap.SmallGroup(288,635);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,2523,514,360,6304,3476,102,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=c^-1,d*c*d^-1=b^-1*c>;
// generators/relations
Export
Subgroup lattice of C3×C42⋊C6 in TeX
Character table of C3×C42⋊C6 in TeX