Aliases: C42⋊4C4⋊C3, C4.1(C42⋊C3), (C22×C4).8A4, C23.10(C2×A4), C22.1(C4.A4), C2.C42.3C6, C23.3A4.3C2, C2.3(C2×C42⋊C3), SmallGroup(192,190)
Series: Derived ►Chief ►Lower central ►Upper central
| C2.C42 — C42⋊4C4⋊C3 |
Generators and relations for C42⋊4C4⋊C3
G = < a,b,c,d | a4=b4=c4=d3=1, ab=ba, cac-1=ab2, dad-1=a2c-1, bc=cb, dbd-1=a2b-1, dcd-1=a-1b2c >
Character table of C42⋊4C4⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ7 | 2 | -2 | 2 | -2 | -1 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -i | -i | i | i | complex lifted from C4.A4 |
| ρ8 | 2 | -2 | 2 | -2 | -1 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | i | i | -i | -i | complex lifted from C4.A4 |
| ρ9 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | complex lifted from C4.A4 |
| ρ10 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | complex lifted from C4.A4 |
| ρ11 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | complex lifted from C4.A4 |
| ρ12 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | complex lifted from C4.A4 |
| ρ13 | 3 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | -1 | -1 | -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 | -3 | -3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1-2i | -1+2i | 1 | 1 | -1-2i | 1 | 1 | -1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ16 | 3 | 3 | -1 | -1 | 0 | 0 | -3 | -3 | 1 | 1 | 1 | 1 | -1+2i | -1-2i | -1 | 1-2i | 1+2i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ17 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | 1 | 1 | -1+2i | -1-2i | 1 | -1+2i | -1-2i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ18 | 3 | 3 | -1 | -1 | 0 | 0 | -3 | -3 | 1 | 1 | 1 | 1 | -1-2i | -1+2i | -1 | 1+2i | 1-2i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ19 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1+2i | -1-2i | 1 | 1 | -1+2i | 1 | 1 | -1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ20 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | 1 | 1 | -1-2i | -1+2i | 1 | -1-2i | -1+2i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ21 | 3 | 3 | -1 | -1 | 0 | 0 | -3 | -3 | 1 | 1 | -1+2i | -1-2i | 1 | 1 | 1-2i | -1 | -1 | 1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ22 | 3 | 3 | -1 | -1 | 0 | 0 | -3 | -3 | 1 | 1 | -1-2i | -1+2i | 1 | 1 | 1+2i | -1 | -1 | 1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ23 | 6 | -6 | -2 | 2 | 0 | 0 | 6i | -6i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 6 | -6 | -2 | 2 | 0 | 0 | -6i | 6i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T301
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 4 9 6)(2 3 10 5)(7 12 14 16)(8 11 13 15)(17 18 19 20)(21 24 23 22)
(1 3)(2 6)(4 10)(5 9)(7 16 14 12)(8 11 13 15)(17 24)(18 23)(19 22)(20 21)
(1 7 22)(2 11 17)(3 8 18)(4 16 21)(5 13 20)(6 12 23)(9 14 24)(10 15 19)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,4,9,6)(2,3,10,5)(7,12,14,16)(8,11,13,15)(17,18,19,20)(21,24,23,22), (1,3)(2,6)(4,10)(5,9)(7,16,14,12)(8,11,13,15)(17,24)(18,23)(19,22)(20,21), (1,7,22)(2,11,17)(3,8,18)(4,16,21)(5,13,20)(6,12,23)(9,14,24)(10,15,19)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,4,9,6)(2,3,10,5)(7,12,14,16)(8,11,13,15)(17,18,19,20)(21,24,23,22), (1,3)(2,6)(4,10)(5,9)(7,16,14,12)(8,11,13,15)(17,24)(18,23)(19,22)(20,21), (1,7,22)(2,11,17)(3,8,18)(4,16,21)(5,13,20)(6,12,23)(9,14,24)(10,15,19) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,4,9,6),(2,3,10,5),(7,12,14,16),(8,11,13,15),(17,18,19,20),(21,24,23,22)], [(1,3),(2,6),(4,10),(5,9),(7,16,14,12),(8,11,13,15),(17,24),(18,23),(19,22),(20,21)], [(1,7,22),(2,11,17),(3,8,18),(4,16,21),(5,13,20),(6,12,23),(9,14,24),(10,15,19)]])
G:=TransitiveGroup(24,301);
►On 24 points - transitive group 24T309
Generators in S24
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 3 4 2)(5 12 6 11)(7 15 13 10)(8 16 14 9)(17 22 19 24)(18 23 20 21)
(1 5)(2 11)(3 12)(4 6)(7 8 13 14)(9 15 16 10)(17 19)(22 24)
(1 14 22)(2 9 17)(3 16 19)(4 8 24)(5 13 23)(6 7 21)(10 18 11)(12 15 20)
G:=sub<Sym(24)| (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,3,4,2)(5,12,6,11)(7,15,13,10)(8,16,14,9)(17,22,19,24)(18,23,20,21), (1,5)(2,11)(3,12)(4,6)(7,8,13,14)(9,15,16,10)(17,19)(22,24), (1,14,22)(2,9,17)(3,16,19)(4,8,24)(5,13,23)(6,7,21)(10,18,11)(12,15,20)>;
G:=Group( (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,3,4,2)(5,12,6,11)(7,15,13,10)(8,16,14,9)(17,22,19,24)(18,23,20,21), (1,5)(2,11)(3,12)(4,6)(7,8,13,14)(9,15,16,10)(17,19)(22,24), (1,14,22)(2,9,17)(3,16,19)(4,8,24)(5,13,23)(6,7,21)(10,18,11)(12,15,20) );
G=PermutationGroup([[(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,3,4,2),(5,12,6,11),(7,15,13,10),(8,16,14,9),(17,22,19,24),(18,23,20,21)], [(1,5),(2,11),(3,12),(4,6),(7,8,13,14),(9,15,16,10),(17,19),(22,24)], [(1,14,22),(2,9,17),(3,16,19),(4,8,24),(5,13,23),(6,7,21),(10,18,11),(12,15,20)]])
G:=TransitiveGroup(24,309);
Matrix representation of C42⋊4C4⋊C3 ►in GL5(𝔽13)
| 9 | 3 | 0 | 0 | 0 |
| 3 | 4 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 10 | 9 | 0 | 0 | 0 |
| 9 | 3 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 9 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
G:=sub<GL(5,GF(13))| [9,3,0,0,0,3,4,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,1],[5,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1],[10,9,0,0,0,9,3,0,0,0,0,0,8,0,0,0,0,0,12,0,0,0,0,0,8],[1,9,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,3,0,0,3,0,0] >;
C42⋊4C4⋊C3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4C_4\rtimes C_3
% in TeX
G:=Group("C4^2:4C4:C3"); // GroupNames label
G:=SmallGroup(192,190);
// by ID
G=gap.SmallGroup(192,190);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,268,934,521,80,2531,3540]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^3=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^2*c^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*b^2*c>;
// generators/relations
Export
Subgroup lattice of C42⋊4C4⋊C3 in TeX
Character table of C42⋊4C4⋊C3 in TeX