non-abelian, soluble, monomial
Aliases: C2≀A4, C24⋊1A4, 2+ 1+4⋊1C6, C2≀C22⋊C3, C23⋊A4⋊1C2, C23.1(C2×A4), C2.4(C24⋊C6), SmallGroup(192,201)
Series: Derived ►Chief ►Lower central ►Upper central
| 2+ 1+4 — C2≀A4 |
Generators and relations for C2≀A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g3=1, ab=ba, faf=ac=ca, ad=da, eae=abd, ag=ga, bc=cb, fbf=gcg-1=bd=db, be=eb, gbg-1=bcd, ece=cd=dc, cf=fc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >
Subgroups: 351 in 58 conjugacy classes, 8 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C23, A4, C2×C6, C22⋊C4, C2×D4, C4○D4, C24, SL2(𝔽3), C2×A4, C23⋊C4, C22≀C2, 2+ 1+4, C22×A4, C2≀C22, C23⋊A4, C2≀A4
Quotients: C1, C2, C3, C6, A4, C2×A4, C24⋊C6, C2≀A4
Character table of C2≀A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 4 | 4 | 6 | 12 | 16 | 16 | 12 | 24 | 16 | 16 | 16 | 16 | 16 | 16 | |
| 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 | ζ32 | ζ3 | 1 | -1 | ζ65 | ζ32 | ζ6 | ζ6 | ζ65 | ζ3 | linear of order 6 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | ζ6 | ζ3 | ζ65 | ζ65 | ζ6 | ζ32 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ7 | 3 | 3 | -3 | -3 | 3 | -1 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ8 | 3 | 3 | 3 | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 4 | -4 | 2 | -2 | 0 | 0 | 1 | 1 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | -1 | orthogonal faithful |
| ρ10 | 4 | -4 | -2 | 2 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | -1 | -1 | 1 | -1 | -1 | orthogonal faithful |
| ρ11 | 4 | -4 | 2 | -2 | 0 | 0 | ζ32 | ζ3 | 0 | 0 | ζ65 | ζ6 | ζ32 | ζ6 | ζ3 | ζ65 | complex faithful |
| ρ12 | 4 | -4 | -2 | 2 | 0 | 0 | ζ32 | ζ3 | 0 | 0 | ζ3 | ζ6 | ζ6 | ζ32 | ζ65 | ζ65 | complex faithful |
| ρ13 | 4 | -4 | 2 | -2 | 0 | 0 | ζ3 | ζ32 | 0 | 0 | ζ6 | ζ65 | ζ3 | ζ65 | ζ32 | ζ6 | complex faithful |
| ρ14 | 4 | -4 | -2 | 2 | 0 | 0 | ζ3 | ζ32 | 0 | 0 | ζ32 | ζ65 | ζ65 | ζ3 | ζ6 | ζ6 | complex faithful |
| ρ15 | 6 | 6 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C6 |
| ρ16 | 6 | 6 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C6 |
►On 8 points - transitive group 8T38
Generators in S8
(1 2)
(3 6)(4 7)
(1 2)(3 6)
(1 2)(3 6)(4 7)(5 8)
(1 5)(2 8)(3 4)(6 7)
(1 3)(2 6)(4 5)(7 8)
(3 4 5)(6 7 8)
G:=sub<Sym(8)| (1,2), (3,6)(4,7), (1,2)(3,6), (1,2)(3,6)(4,7)(5,8), (1,5)(2,8)(3,4)(6,7), (1,3)(2,6)(4,5)(7,8), (3,4,5)(6,7,8)>;
G:=Group( (1,2), (3,6)(4,7), (1,2)(3,6), (1,2)(3,6)(4,7)(5,8), (1,5)(2,8)(3,4)(6,7), (1,3)(2,6)(4,5)(7,8), (3,4,5)(6,7,8) );
G=PermutationGroup([[(1,2)], [(3,6),(4,7)], [(1,2),(3,6)], [(1,2),(3,6),(4,7),(5,8)], [(1,5),(2,8),(3,4),(6,7)], [(1,3),(2,6),(4,5),(7,8)], [(3,4,5),(6,7,8)]])
G:=TransitiveGroup(8,38);
►On 16 points - transitive group 16T425
Generators in S16
(1 4)(2 3)(5 9)(6 10)(7 8)(11 14)(12 15)(13 16)
(1 12)(2 10)(3 6)(4 15)(5 7)(8 9)(11 13)(14 16)
(1 8)(2 13)(3 16)(4 7)(5 15)(6 14)(9 12)(10 11)
(1 2)(3 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)
(3 15)(4 6)(5 7)(8 13)(9 11)(14 16)
(3 16)(4 7)(5 6)(9 11)(10 12)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
G:=sub<Sym(16)| (1,4)(2,3)(5,9)(6,10)(7,8)(11,14)(12,15)(13,16), (1,12)(2,10)(3,6)(4,15)(5,7)(8,9)(11,13)(14,16), (1,8)(2,13)(3,16)(4,7)(5,15)(6,14)(9,12)(10,11), (1,2)(3,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (3,15)(4,6)(5,7)(8,13)(9,11)(14,16), (3,16)(4,7)(5,6)(9,11)(10,12)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16)>;
G:=Group( (1,4)(2,3)(5,9)(6,10)(7,8)(11,14)(12,15)(13,16), (1,12)(2,10)(3,6)(4,15)(5,7)(8,9)(11,13)(14,16), (1,8)(2,13)(3,16)(4,7)(5,15)(6,14)(9,12)(10,11), (1,2)(3,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (3,15)(4,6)(5,7)(8,13)(9,11)(14,16), (3,16)(4,7)(5,6)(9,11)(10,12)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16) );
G=PermutationGroup([[(1,4),(2,3),(5,9),(6,10),(7,8),(11,14),(12,15),(13,16)], [(1,12),(2,10),(3,6),(4,15),(5,7),(8,9),(11,13),(14,16)], [(1,8),(2,13),(3,16),(4,7),(5,15),(6,14),(9,12),(10,11)], [(1,2),(3,4),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12)], [(3,15),(4,6),(5,7),(8,13),(9,11),(14,16)], [(3,16),(4,7),(5,6),(9,11),(10,12),(14,15)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)]])
G:=TransitiveGroup(16,425);
►On 16 points - transitive group 16T427
Generators in S16
(1 3)(2 4)(5 12)(6 13)(7 11)(8 15)(9 16)(10 14)
(6 15)(7 16)(8 13)(9 11)
(1 4)(2 3)(6 15)(8 13)
(1 4)(2 3)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)
(1 14)(2 10)(3 12)(4 5)(6 7)(8 9)(11 13)(15 16)
(1 15)(2 8)(3 13)(4 6)(5 7)(9 10)(11 12)(14 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
G:=sub<Sym(16)| (1,3)(2,4)(5,12)(6,13)(7,11)(8,15)(9,16)(10,14), (6,15)(7,16)(8,13)(9,11), (1,4)(2,3)(6,15)(8,13), (1,4)(2,3)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (1,14)(2,10)(3,12)(4,5)(6,7)(8,9)(11,13)(15,16), (1,15)(2,8)(3,13)(4,6)(5,7)(9,10)(11,12)(14,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16)>;
G:=Group( (1,3)(2,4)(5,12)(6,13)(7,11)(8,15)(9,16)(10,14), (6,15)(7,16)(8,13)(9,11), (1,4)(2,3)(6,15)(8,13), (1,4)(2,3)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (1,14)(2,10)(3,12)(4,5)(6,7)(8,9)(11,13)(15,16), (1,15)(2,8)(3,13)(4,6)(5,7)(9,10)(11,12)(14,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16) );
G=PermutationGroup([[(1,3),(2,4),(5,12),(6,13),(7,11),(8,15),(9,16),(10,14)], [(6,15),(7,16),(8,13),(9,11)], [(1,4),(2,3),(6,15),(8,13)], [(1,4),(2,3),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12)], [(1,14),(2,10),(3,12),(4,5),(6,7),(8,9),(11,13),(15,16)], [(1,15),(2,8),(3,13),(4,6),(5,7),(9,10),(11,12),(14,16)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)]])
G:=TransitiveGroup(16,427);
►On 24 points - transitive group 24T288
Generators in S24
(1 23)(2 24)(3 22)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(2 24)(3 22)(11 13)(12 14)(16 19)(17 20)
(2 24)(4 8)(5 9)(6 7)(11 13)(16 19)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(1 8)(2 20)(3 11)(4 23)(5 15)(6 18)(7 21)(9 10)(12 19)(13 22)(14 16)(17 24)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 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)| (1,23)(2,24)(3,22)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,24)(3,22)(11,13)(12,14)(16,19)(17,20), (2,24)(4,8)(5,9)(6,7)(11,13)(16,19), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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( (1,23)(2,24)(3,22)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,24)(3,22)(11,13)(12,14)(16,19)(17,20), (2,24)(4,8)(5,9)(6,7)(11,13)(16,19), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (1,8)(2,20)(3,11)(4,23)(5,15)(6,18)(7,21)(9,10)(12,19)(13,22)(14,16)(17,24), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (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([[(1,23),(2,24),(3,22),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(2,24),(3,22),(11,13),(12,14),(16,19),(17,20)], [(2,24),(4,8),(5,9),(6,7),(11,13),(16,19)], [(1,23),(2,24),(3,22),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(1,8),(2,20),(3,11),(4,23),(5,15),(6,18),(7,21),(9,10),(12,19),(13,22),(14,16),(17,24)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(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(24,288);
►On 24 points - transitive group 24T425
Generators in S24
(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)
(1 23)(2 10)(3 13)(5 20)(6 21)(7 18)(9 17)(11 22)(12 14)(15 24)
(1 12)(3 13)(4 16)(5 9)(6 18)(7 21)(8 19)(11 22)(14 23)(17 20)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(2 9)(3 7)(5 24)(6 22)(10 17)(11 21)(12 14)(13 18)(15 20)(16 19)
(1 8)(3 7)(4 23)(6 22)(10 15)(11 18)(12 19)(13 21)(14 16)(17 20)
(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)| (4,16)(5,17)(6,18)(7,21)(8,19)(9,20), (1,23)(2,10)(3,13)(5,20)(6,21)(7,18)(9,17)(11,22)(12,14)(15,24), (1,12)(3,13)(4,16)(5,9)(6,18)(7,21)(8,19)(11,22)(14,23)(17,20), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,9)(3,7)(5,24)(6,22)(10,17)(11,21)(12,14)(13,18)(15,20)(16,19), (1,8)(3,7)(4,23)(6,22)(10,15)(11,18)(12,19)(13,21)(14,16)(17,20), (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( (4,16)(5,17)(6,18)(7,21)(8,19)(9,20), (1,23)(2,10)(3,13)(5,20)(6,21)(7,18)(9,17)(11,22)(12,14)(15,24), (1,12)(3,13)(4,16)(5,9)(6,18)(7,21)(8,19)(11,22)(14,23)(17,20), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (2,9)(3,7)(5,24)(6,22)(10,17)(11,21)(12,14)(13,18)(15,20)(16,19), (1,8)(3,7)(4,23)(6,22)(10,15)(11,18)(12,19)(13,21)(14,16)(17,20), (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([[(4,16),(5,17),(6,18),(7,21),(8,19),(9,20)], [(1,23),(2,10),(3,13),(5,20),(6,21),(7,18),(9,17),(11,22),(12,14),(15,24)], [(1,12),(3,13),(4,16),(5,9),(6,18),(7,21),(8,19),(11,22),(14,23),(17,20)], [(1,23),(2,24),(3,22),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(2,9),(3,7),(5,24),(6,22),(10,17),(11,21),(12,14),(13,18),(15,20),(16,19)], [(1,8),(3,7),(4,23),(6,22),(10,15),(11,18),(12,19),(13,21),(14,16),(17,20)], [(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(24,425);
Polynomial with Galois group C2≀A4 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 8T38 | x8-3x7-13x6+18x5+42x4-17x3-31x2+2x+4 | 22·89·1634·6532 |
Matrix representation of C2≀A4 ►in GL4(ℤ) generated by
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0] >;
C2≀A4 in GAP, Magma, Sage, TeX
C_2\wr A_4
% in TeX
G:=Group("C2wrA4"); // GroupNames label
G:=SmallGroup(192,201);
// by ID
G=gap.SmallGroup(192,201);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,1683,262,851,375,3540,1027]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^3=1,a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,e*a*e=a*b*d,a*g=g*a,b*c=c*b,f*b*f=g*c*g^-1=b*d=d*b,b*e=e*b,g*b*g^-1=b*c*d,e*c*e=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations
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