Aliases: C42⋊1C12, C42⋊5C4⋊C3, C42⋊C3⋊2C4, C22.3(C4×A4), (C2×C42).1C6, (C22×C4).1A4, C23.12(C2×A4), C2.1(C42⋊C6), (C2×C42⋊C3).1C2, SmallGroup(192,192)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊C12 |
Generators and relations for C42⋊C12
G = < a,b,c | a4=b4=c12=1, ab=ba, cac-1=a-1b-1, cbc-1=a-1 >
Character table of C42⋊C12
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -1 | 1 | -1 | 1 | i | -i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | -1 | 1 | -1 | 1 | -i | i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | -i | i | -1 | 1 | -1 | 1 | i | -i | ζ65 | ζ6 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | linear of order 12 |
| ρ10 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | -i | i | -1 | 1 | -1 | 1 | i | -i | ζ6 | ζ65 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | linear of order 12 |
| ρ11 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | i | -i | -1 | 1 | -1 | 1 | -i | i | ζ65 | ζ6 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | i | -i | -1 | 1 | -1 | 1 | -i | i | ζ6 | ζ65 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | linear of order 12 |
| ρ13 | 3 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ14 | 3 | 3 | 3 | 3 | 0 | 0 | -3 | -3 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | -3 | 3 | -3 | 0 | 0 | 3i | -3i | 1 | -1 | 1 | -1 | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×A4 |
| ρ16 | 3 | -3 | 3 | -3 | 0 | 0 | -3i | 3i | 1 | -1 | 1 | -1 | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×A4 |
| ρ17 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
| ρ18 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
►On 24 points - transitive group 24T299
Generators in S24
(1 4 7 10)(2 22 8 16)(3 14)(5 19 11 13)(6 23)(9 20)(12 17)(15 24 21 18)
(1 18)(2 11 8 5)(3 17 9 23)(4 15)(6 14 12 20)(7 24)(10 21)(13 16 19 22)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,4,7,10)(2,22,8,16)(3,14)(5,19,11,13)(6,23)(9,20)(12,17)(15,24,21,18), (1,18)(2,11,8,5)(3,17,9,23)(4,15)(6,14,12,20)(7,24)(10,21)(13,16,19,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,4,7,10)(2,22,8,16)(3,14)(5,19,11,13)(6,23)(9,20)(12,17)(15,24,21,18), (1,18)(2,11,8,5)(3,17,9,23)(4,15)(6,14,12,20)(7,24)(10,21)(13,16,19,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,4,7,10),(2,22,8,16),(3,14),(5,19,11,13),(6,23),(9,20),(12,17),(15,24,21,18)], [(1,18),(2,11,8,5),(3,17,9,23),(4,15),(6,14,12,20),(7,24),(10,21),(13,16,19,22)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,299);
►On 24 points - transitive group 24T305
Generators in S24
(1 10)(2 22 8 16)(3 14 9 20)(4 7)(5 13 11 19)(6 23 12 17)(15 24)(18 21)
(1 24 7 18)(2 11)(3 17 9 23)(4 21 10 15)(5 8)(6 20 12 14)(13 16)(19 22)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,10)(2,22,8,16)(3,14,9,20)(4,7)(5,13,11,19)(6,23,12,17)(15,24)(18,21), (1,24,7,18)(2,11)(3,17,9,23)(4,21,10,15)(5,8)(6,20,12,14)(13,16)(19,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,10)(2,22,8,16)(3,14,9,20)(4,7)(5,13,11,19)(6,23,12,17)(15,24)(18,21), (1,24,7,18)(2,11)(3,17,9,23)(4,21,10,15)(5,8)(6,20,12,14)(13,16)(19,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,10),(2,22,8,16),(3,14,9,20),(4,7),(5,13,11,19),(6,23,12,17),(15,24),(18,21)], [(1,24,7,18),(2,11),(3,17,9,23),(4,21,10,15),(5,8),(6,20,12,14),(13,16),(19,22)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,305);
►On 24 points - transitive group 24T308
Generators in S24
(2 13 8 19)(3 20 9 14)(4 10)(5 16 11 22)(6 17 12 23)(15 21)
(1 18 7 24)(3 20 9 14)(4 15 10 21)(5 11)(6 23 12 17)(16 22)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (2,13,8,19)(3,20,9,14)(4,10)(5,16,11,22)(6,17,12,23)(15,21), (1,18,7,24)(3,20,9,14)(4,15,10,21)(5,11)(6,23,12,17)(16,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (2,13,8,19)(3,20,9,14)(4,10)(5,16,11,22)(6,17,12,23)(15,21), (1,18,7,24)(3,20,9,14)(4,15,10,21)(5,11)(6,23,12,17)(16,22), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(2,13,8,19),(3,20,9,14),(4,10),(5,16,11,22),(6,17,12,23),(15,21)], [(1,18,7,24),(3,20,9,14),(4,15,10,21),(5,11),(6,23,12,17),(16,22)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,308);
Matrix representation of C42⋊C12 ►in GL6(𝔽13)
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [0,5,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,5,0],[0,0,0,0,0,5,0,0,0,0,5,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0] >;
C42⋊C12 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_{12} % in TeX
G:=Group("C4^2:C12"); // GroupNames label
G:=SmallGroup(192,192);
// by ID
G=gap.SmallGroup(192,192);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,1683,346,360,4204,2321,102,2028,3541]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C42⋊C12 in TeX
Character table of C42⋊C12 in TeX