direct product, metabelian, soluble, monomial, A-group
Aliases: C4×A4, C22⋊C12, C23.C6, (C22×C4)⋊C3, C2.1(C2×A4), (C2×A4).2C2, SmallGroup(48,31)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C4×A4 |
Generators and relations for C4×A4
G = < a,b,c,d | a4=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Character table of C4×A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 1 | 1 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -i | i | -i | i | ζ65 | ζ6 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | i | -i | i | -i | ζ6 | ζ65 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | i | -i | i | -i | ζ65 | ζ6 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -i | i | -i | i | ζ6 | ζ65 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | linear of order 12 |
| ρ13 | 3 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ14 | 3 | 3 | -1 | -1 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | -3 | 1 | -1 | 0 | 0 | -3i | 3i | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 3 | -3 | 1 | -1 | 0 | 0 | 3i | -3i | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 12 points - transitive group 12T29
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(5 7)(6 8)(9 11)(10 12)
(1 3)(2 4)(9 11)(10 12)
(1 7 11)(2 8 12)(3 5 9)(4 6 10)
G:=sub<Sym(12)| (1,2,3,4)(5,6,7,8)(9,10,11,12), (5,7)(6,8)(9,11)(10,12), (1,3)(2,4)(9,11)(10,12), (1,7,11)(2,8,12)(3,5,9)(4,6,10)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12), (5,7)(6,8)(9,11)(10,12), (1,3)(2,4)(9,11)(10,12), (1,7,11)(2,8,12)(3,5,9)(4,6,10) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(5,7),(6,8),(9,11),(10,12)], [(1,3),(2,4),(9,11),(10,12)], [(1,7,11),(2,8,12),(3,5,9),(4,6,10)]])
G:=TransitiveGroup(12,29);
►On 16 points - transitive group 16T57
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)
(5 15 9)(6 16 10)(7 13 11)(8 14 12)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (5,15,9)(6,16,10)(7,13,11)(8,14,12)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14), (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (5,15,9)(6,16,10)(7,13,11)(8,14,12) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10),(2,11),(3,12),(4,9),(5,15),(6,16),(7,13),(8,14)], [(1,6),(2,7),(3,8),(4,5),(9,15),(10,16),(11,13),(12,14)], [(5,15,9),(6,16,10),(7,13,11),(8,14,12)]])
G:=TransitiveGroup(16,57);
►On 24 points - transitive group 24T55
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 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 23)(18 24)(19 21)(20 22)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
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), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24)>;
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), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,23)(18,24)(19,21)(20,22), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24) );
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)], [(1,3),(2,4),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12),(17,21),(18,22),(19,23),(20,24)], [(1,11),(2,12),(3,9),(4,10),(5,7),(6,8),(13,15),(14,16),(17,23),(18,24),(19,21),(20,22)], [(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,21),(10,13,22),(11,14,23),(12,15,24)]])
G:=TransitiveGroup(24,55);
►On 24 points - transitive group 24T56
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)
(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 9)(4 10)(17 21)(18 22)(19 23)(20 24)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
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), (5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24)>;
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), (5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,11)(2,12)(3,9)(4,10)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24) );
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)], [(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(1,11),(2,12),(3,9),(4,10),(17,21),(18,22),(19,23),(20,24)], [(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,21),(10,13,22),(11,14,23),(12,15,24)]])
G:=TransitiveGroup(24,56);
C4×A4 is a maximal subgroup of
A4⋊C8 A4⋊Q8 C4⋊S4 C23.SL2(𝔽3) C42⋊4C4⋊C3 C24⋊C12 C42⋊C12 C42⋊2C12 C23⋊2D4⋊C3 (C22×C4).A4 C23.19(C2×A4) 2+ 1+4.C6 C4○D4⋊A4 2+ 1+4.3C6 Dic7⋊A4
C4×A4 is a maximal quotient of
C8.A4 C24⋊C12 C42⋊C12 C42⋊2C12 Dic7⋊A4
| action | f(x) | Disc(f) |
|---|---|---|
| 12T29 | x12-26x10+195x8-663x6+1144x4-975x2+325 | 212·514·1311 |
Matrix representation of C4×A4 ►in GL3(𝔽5) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 3 | 0 |
| 0 | 0 | 4 |
| 3 | 0 | 0 |
G:=sub<GL(3,GF(5))| [3,0,0,0,3,0,0,0,3],[4,0,0,0,1,0,0,0,4],[1,0,0,0,4,0,0,0,4],[0,0,3,3,0,0,0,4,0] >;
C4×A4 in GAP, Magma, Sage, TeX
C_4\times A_4
% in TeX
G:=Group("C4xA4"); // GroupNames label
G:=SmallGroup(48,31);
// by ID
G=gap.SmallGroup(48,31);
# by ID
G:=PCGroup([5,-2,-3,-2,-2,2,30,248,459]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C4×A4 in TeX
Character table of C4×A4 in TeX