direct product, metabelian, soluble, monomial, A-group
Aliases: C22×A4, C23⋊C6, C24⋊1C3, C22⋊(C2×C6), SmallGroup(48,49)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C22×A4 |
Generators and relations for C22×A4
G = < a,b,c,d,e | a2=b2=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
3C2
3C2
3C2
3C2
4C3
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
3C22
4C6
4C6
4C6
3C23
3C23
3C23
3C23
Character table of C22×A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 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 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | linear of order 6 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ12 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | linear of order 6 |
| ρ13 | 3 | -3 | -3 | 3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | -3 | 3 | -3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ16 | 3 | 3 | -3 | -3 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
►On 12 points - transitive group 12T25
Generators in S12
(1 4)(2 5)(3 6)(7 10)(8 11)(9 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:=sub<Sym(12)| (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (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( (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (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=PermutationGroup([[(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(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:=TransitiveGroup(12,25);
►On 12 points - transitive group 12T26
Generators in S12
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(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)| (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (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( (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (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([[(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(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(12,26);
►On 16 points - transitive group 16T58
Generators in S16
(1 3)(2 4)(5 8)(6 9)(7 10)(11 14)(12 15)(13 16)
(1 2)(3 4)(5 16)(6 14)(7 15)(8 13)(9 11)(10 12)
(1 13)(2 8)(3 16)(4 5)(6 7)(9 10)(11 12)(14 15)
(1 11)(2 9)(3 14)(4 6)(5 7)(8 10)(12 13)(15 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
G:=sub<Sym(16)| (1,3)(2,4)(5,8)(6,9)(7,10)(11,14)(12,15)(13,16), (1,2)(3,4)(5,16)(6,14)(7,15)(8,13)(9,11)(10,12), (1,13)(2,8)(3,16)(4,5)(6,7)(9,10)(11,12)(14,15), (1,11)(2,9)(3,14)(4,6)(5,7)(8,10)(12,13)(15,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16)>;
G:=Group( (1,3)(2,4)(5,8)(6,9)(7,10)(11,14)(12,15)(13,16), (1,2)(3,4)(5,16)(6,14)(7,15)(8,13)(9,11)(10,12), (1,13)(2,8)(3,16)(4,5)(6,7)(9,10)(11,12)(14,15), (1,11)(2,9)(3,14)(4,6)(5,7)(8,10)(12,13)(15,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16) );
G=PermutationGroup([[(1,3),(2,4),(5,8),(6,9),(7,10),(11,14),(12,15),(13,16)], [(1,2),(3,4),(5,16),(6,14),(7,15),(8,13),(9,11),(10,12)], [(1,13),(2,8),(3,16),(4,5),(6,7),(9,10),(11,12),(14,15)], [(1,11),(2,9),(3,14),(4,6),(5,7),(8,10),(12,13),(15,16)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)]])
G:=TransitiveGroup(16,58);
►On 24 points - transitive group 24T49
Generators in S24
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(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)| (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (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( (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (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([[(1,15),(2,13),(3,14),(4,20),(5,21),(6,19),(7,24),(8,22),(9,23),(10,18),(11,16),(12,17)], [(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(24,49);
►On 24 points - transitive group 24T50
Generators in S24
(1 22)(2 23)(3 24)(4 7)(5 8)(6 9)(10 16)(11 17)(12 18)(13 19)(14 20)(15 21)
(1 10)(2 11)(3 12)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 8)(3 9)(5 23)(6 24)(11 14)(12 15)(17 20)(18 21)
(1 7)(3 9)(4 22)(6 24)(10 13)(12 15)(16 19)(18 21)
(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)| (1,22)(2,23)(3,24)(4,7)(5,8)(6,9)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21), (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (2,8)(3,9)(5,23)(6,24)(11,14)(12,15)(17,20)(18,21), (1,7)(3,9)(4,22)(6,24)(10,13)(12,15)(16,19)(18,21), (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( (1,22)(2,23)(3,24)(4,7)(5,8)(6,9)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21), (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (2,8)(3,9)(5,23)(6,24)(11,14)(12,15)(17,20)(18,21), (1,7)(3,9)(4,22)(6,24)(10,13)(12,15)(16,19)(18,21), (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([[(1,22),(2,23),(3,24),(4,7),(5,8),(6,9),(10,16),(11,17),(12,18),(13,19),(14,20),(15,21)], [(1,10),(2,11),(3,12),(4,19),(5,20),(6,21),(7,13),(8,14),(9,15),(16,22),(17,23),(18,24)], [(2,8),(3,9),(5,23),(6,24),(11,14),(12,15),(17,20),(18,21)], [(1,7),(3,9),(4,22),(6,24),(10,13),(12,15),(16,19),(18,21)], [(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(24,50);
C22×A4 is a maximal subgroup of
A4⋊D4 C24.A4 C24.2A4 C24.3A4 C2≀A4 C24.6A4 C24⋊A4 C24.7A4 C24⋊5A4
C22×A4 is a maximal quotient of
Q8.A4 D4.A4 C24.6A4 C24⋊A4
| action | f(x) | Disc(f) |
|---|---|---|
| 12T25 | x12-12x10+48x8-77x6+48x4-12x2+1 | 212·318·1992 |
| 12T26 | x12+2x10-5x8-4x6-5x4+2x2+1 | 248·78 |
Matrix representation of C22×A4 ►in GL4(𝔽7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 6 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,1],[1,0,0,0,0,6,0,0,0,0,1,0,0,0,0,6],[4,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C22×A4 in GAP, Magma, Sage, TeX
C_2^2\times A_4
% in TeX
G:=Group("C2^2xA4"); // GroupNames label
G:=SmallGroup(48,49);
// by ID
G=gap.SmallGroup(48,49);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,2,133,239]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of C22×A4 in TeX
Character table of C22×A4 in TeX