direct product, metabelian, soluble, monomial, A-group
Aliases: C2×A4, C2≀C3, AΣL1(𝔽8), C23⋊C3, C22⋊C6, SmallGroup(24,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C2×A4 |
Generators and relations for C2×A4
G = < a,b,c,d | a2=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Character table of C2×A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 6 points - transitive group 6T6
Generators in S6
(1 4)(2 5)(3 6)
(2 5)(3 6)
(1 4)(3 6)
(1 2 3)(4 5 6)
G:=sub<Sym(6)| (1,4)(2,5)(3,6), (2,5)(3,6), (1,4)(3,6), (1,2,3)(4,5,6)>;
G:=Group( (1,4)(2,5)(3,6), (2,5)(3,6), (1,4)(3,6), (1,2,3)(4,5,6) );
G=PermutationGroup([[(1,4),(2,5),(3,6)], [(2,5),(3,6)], [(1,4),(3,6)], [(1,2,3),(4,5,6)]])
G:=TransitiveGroup(6,6);
►On 8 points - transitive group 8T13
Generators in S8
(1 2)(3 6)(4 7)(5 8)
(1 6)(2 3)(4 5)(7 8)
(1 7)(2 4)(3 5)(6 8)
(3 4 5)(6 7 8)
G:=sub<Sym(8)| (1,2)(3,6)(4,7)(5,8), (1,6)(2,3)(4,5)(7,8), (1,7)(2,4)(3,5)(6,8), (3,4,5)(6,7,8)>;
G:=Group( (1,2)(3,6)(4,7)(5,8), (1,6)(2,3)(4,5)(7,8), (1,7)(2,4)(3,5)(6,8), (3,4,5)(6,7,8) );
G=PermutationGroup([[(1,2),(3,6),(4,7),(5,8)], [(1,6),(2,3),(4,5),(7,8)], [(1,7),(2,4),(3,5),(6,8)], [(3,4,5),(6,7,8)]])
G:=TransitiveGroup(8,13);
►On 12 points - transitive group 12T6
Generators in S12
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 4)(2 9)(3 12)(5 11)(6 7)(8 10)
(1 10)(2 5)(3 7)(4 8)(6 12)(9 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
G:=sub<Sym(12)| (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,4)(2,9)(3,12)(5,11)(6,7)(8,10), (1,10)(2,5)(3,7)(4,8)(6,12)(9,11), (1,2,3)(4,5,6)(7,8,9)(10,11,12)>;
G:=Group( (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,4)(2,9)(3,12)(5,11)(6,7)(8,10), (1,10)(2,5)(3,7)(4,8)(6,12)(9,11), (1,2,3)(4,5,6)(7,8,9)(10,11,12) );
G=PermutationGroup([[(1,4),(2,5),(3,6),(7,12),(8,10),(9,11)], [(1,4),(2,9),(3,12),(5,11),(6,7),(8,10)], [(1,10),(2,5),(3,7),(4,8),(6,12),(9,11)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]])
G:=TransitiveGroup(12,6);
►On 12 points - transitive group 12T7
Generators in S12
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)
(2 8)(3 9)(5 11)(6 12)
(1 7)(3 9)(4 10)(6 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
G:=sub<Sym(12)| (1,10)(2,11)(3,12)(4,7)(5,8)(6,9), (2,8)(3,9)(5,11)(6,12), (1,7)(3,9)(4,10)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12)>;
G:=Group( (1,10)(2,11)(3,12)(4,7)(5,8)(6,9), (2,8)(3,9)(5,11)(6,12), (1,7)(3,9)(4,10)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,7),(5,8),(6,9)], [(2,8),(3,9),(5,11),(6,12)], [(1,7),(3,9),(4,10),(6,12)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]])
G:=TransitiveGroup(12,7);
►Regular action on 24 points - transitive group 24T9
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 15)(2 16)(3 12)(4 20)(5 9)(6 24)(7 19)(8 22)(10 18)(11 13)(14 17)(21 23)
(1 10)(2 13)(3 17)(4 22)(5 21)(6 7)(8 20)(9 23)(11 16)(12 14)(15 18)(19 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)
G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,16)(3,12)(4,20)(5,9)(6,24)(7,19)(8,22)(10,18)(11,13)(14,17)(21,23), (1,10)(2,13)(3,17)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,14)(15,18)(19,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)>;
G:=Group( (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,16)(3,12)(4,20)(5,9)(6,24)(7,19)(8,22)(10,18)(11,13)(14,17)(21,23), (1,10)(2,13)(3,17)(4,22)(5,21)(6,7)(8,20)(9,23)(11,16)(12,14)(15,18)(19,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24) );
G=PermutationGroup([[(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,15),(2,16),(3,12),(4,20),(5,9),(6,24),(7,19),(8,22),(10,18),(11,13),(14,17),(21,23)], [(1,10),(2,13),(3,17),(4,22),(5,21),(6,7),(8,20),(9,23),(11,16),(12,14),(15,18),(19,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)]])
G:=TransitiveGroup(24,9);
C2×A4 is a maximal subgroup of
A4⋊C4 C23.3A4 C24⋊C6 C42⋊C6 C23.A4 Q8⋊A4 C23⋊A4 AΓL1(𝔽8) D7⋊A4 D13⋊A4 D19⋊A4
C2×A4 is a maximal quotient of
C4.A4 C24⋊C6 C42⋊C6 C23.A4 D7⋊A4 D13⋊A4 D19⋊A4
| action | f(x) | Disc(f) |
|---|---|---|
| 6T6 | x6-x4-2x2+1 | -26·74 |
| 8T13 | x8-x7+x6-x5-x4+x3+x2+x+1 | 74·134 |
| 12T6 | x12-20x10+148x8-503x6+800x4-512x2+64 | 242·114·438 |
| 12T7 | x12-21x10+135x8-345x6+321x4-39x2+1 | 212·316·176·12794 |
Matrix representation of C2×A4 ►in GL3(ℤ) generated by
| -1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | -1 |
| 1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | -1 |
| -1 | 0 | 0 |
| 0 | -1 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,Integers())| [-1,0,0,0,-1,0,0,0,-1],[1,0,0,0,-1,0,0,0,-1],[-1,0,0,0,-1,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;
C2×A4 in GAP, Magma, Sage, TeX
C_2\times A_4
% in TeX
G:=Group("C2xA4"); // GroupNames label
G:=SmallGroup(24,13);
// by ID
G=gap.SmallGroup(24,13);
# by ID
G:=PCGroup([4,-2,-3,-2,2,78,151]);
// Polycyclic
G:=Group<a,b,c,d|a^2=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 C2×A4 in TeX
Character table of C2×A4 in TeX