Aliases: C42⋊2C12, C42⋊9C4⋊C3, C42⋊C3⋊4C4, C22.4(C4×A4), (C2×C42).2C6, (C22×C4).2A4, C23.13(C2×A4), C2.1(C23.A4), (C2×C42⋊C3).2C2, SmallGroup(192,193)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊2C12 |
Generators and relations for C42⋊2C12
G = < a,b,c | a4=b4=c12=1, ab=ba, cac-1=a-1b, cbc-1=ab2 >
Character table of C42⋊2C12
| 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 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.A4 |
| ρ18 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.A4 |
| ρ19 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ20 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T311
Generators in S24
(2 16 8 22)(3 17 9 23)(5 13 11 19)(6 14 12 20)
(1 15 7 21)(2 8)(3 17 9 23)(4 24 10 18)(5 11)(6 14 12 20)(13 19)(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)| (2,16,8,22)(3,17,9,23)(5,13,11,19)(6,14,12,20), (1,15,7,21)(2,8)(3,17,9,23)(4,24,10,18)(5,11)(6,14,12,20)(13,19)(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( (2,16,8,22)(3,17,9,23)(5,13,11,19)(6,14,12,20), (1,15,7,21)(2,8)(3,17,9,23)(4,24,10,18)(5,11)(6,14,12,20)(13,19)(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([[(2,16,8,22),(3,17,9,23),(5,13,11,19),(6,14,12,20)], [(1,15,7,21),(2,8),(3,17,9,23),(4,24,10,18),(5,11),(6,14,12,20),(13,19),(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(24,311);
Matrix representation of C42⋊2C12 ►in GL6(𝔽3)
| 0 | 2 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 2 | 0 |
| 2 | 1 | 2 | 0 | 2 | 2 |
| 2 | 1 | 2 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 2 | 0 |
| 1 | 1 | 0 | 0 | 2 | 0 |
| 0 | 0 | 2 | 0 | 2 | 2 |
| 2 | 0 | 2 | 2 | 0 | 1 |
| 2 | 0 | 2 | 2 | 0 | 2 |
| 2 | 0 | 0 | 1 | 0 | 1 |
| 2 | 0 | 0 | 2 | 0 | 0 |
| 0 | 2 | 0 | 1 | 0 | 2 |
| 1 | 1 | 1 | 2 | 0 | 2 |
| 1 | 2 | 1 | 1 | 2 | 2 |
| 2 | 2 | 1 | 1 | 1 | 2 |
| 2 | 0 | 0 | 2 | 0 | 2 |
| 1 | 2 | 1 | 2 | 0 | 1 |
| 0 | 2 | 2 | 2 | 2 | 0 |
G:=sub<GL(6,GF(3))| [0,1,2,2,1,1,2,0,1,1,1,1,1,1,2,2,1,0,0,0,0,0,1,0,1,2,2,0,2,2,0,0,2,0,0,0],[0,2,2,2,2,0,0,0,0,0,0,2,2,2,2,0,0,0,0,2,2,1,2,1,2,0,0,0,0,0,2,1,2,1,0,2],[1,1,2,2,1,0,1,2,2,0,2,2,1,1,1,0,1,2,2,1,1,2,2,2,0,2,1,0,0,2,2,2,2,2,1,0] >;
C42⋊2C12 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2C_{12} % in TeX
G:=Group("C4^2:2C12"); // GroupNames label
G:=SmallGroup(192,193);
// by ID
G=gap.SmallGroup(192,193);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,4371,346,360,2524,2321,102,2028,3541]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a*b^2>;
// generators/relations
Export
Subgroup lattice of C42⋊2C12 in TeX
Character table of C42⋊2C12 in TeX