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G = A4≀C2order 288 = 25·32

Wreath product of A4 by C2

non-abelian, soluble, monomial

Aliases: A4C2, A421C2, C22⋊A4⋊C6, C22⋊S4⋊C3, C22⋊A41S3, C242(C3×S3), SmallGroup(288,1025)

Series: Derived Chief Lower central Upper central

C1C24C22⋊A4 — A4≀C2
C1C24C22⋊A4A42 — A4≀C2
C22⋊A4 — A4≀C2
C1

Generators and relations for A4≀C2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f6=1, eae-1=ab=ba, ac=ca, faf-1=ad=da, fdf-1=bc=cb, bd=db, ebe-1=a, fbf-1=abc, ece-1=cd=dc, fcf-1=abd, ede-1=c, fef-1=e-1 >

6C2
9C2
12C2
8C3
16C3
16C3
2C22
3C22
3C22
9C22
18C22
18C22
18C4
16S3
24C6
48C6
16C32
3C23
6C23
9C2×C4
9C23
18D4
18D4
2A4
8C2×C6
8A4
8A4
12A4
12A4
16C3×S3
9C2×D4
9C22⋊C4
6C2×A4
12C2×A4
12S4
8C3×A4
3C22≀C2
2C22×A4
3C24⋊C6

Character table of A4≀C2

 class 12A2B2C3A3B3C3D3E46A6B6C6D
 size 16912881616323624244848
ρ111111111111111    trivial
ρ2111-111111-111-1-1    linear of order 2
ρ3111-1ζ32ζ3ζ32ζ31-1ζ3ζ32ζ65ζ6    linear of order 6
ρ41111ζ32ζ3ζ32ζ311ζ3ζ32ζ3ζ32    linear of order 3
ρ5111-1ζ3ζ32ζ3ζ321-1ζ32ζ3ζ6ζ65    linear of order 6
ρ61111ζ3ζ32ζ3ζ3211ζ32ζ3ζ32ζ3    linear of order 3
ρ72220-1-122-10-1-100    orthogonal lifted from S3
ρ82220ζ65ζ6-1+-3-1--3-10ζ6ζ6500    complex lifted from C3×S3
ρ92220ζ6ζ65-1--3-1+-3-10ζ65ζ600    complex lifted from C3×S3
ρ1062-20330000-1-100    orthogonal faithful
ρ1162-20-3-3-3/2-3+3-3/20000ζ65ζ600    complex faithful
ρ1262-20-3+3-3/2-3-3-3/20000ζ6ζ6500    complex faithful
ρ139-31300000-10000    orthogonal faithful
ρ149-31-30000010000    orthogonal faithful

Permutation representations of A4≀C2
On 8 points - transitive group 8T42
Generators in S8
(1 4)(2 3)(5 7)(6 8)
(1 8)(2 5)(3 7)(4 6)
(1 8)(4 6)
(1 6)(4 8)
(3 5 7)(4 8 6)
(1 2)(3 4 5 6 7 8)

G:=sub<Sym(8)| (1,4)(2,3)(5,7)(6,8), (1,8)(2,5)(3,7)(4,6), (1,8)(4,6), (1,6)(4,8), (3,5,7)(4,8,6), (1,2)(3,4,5,6,7,8)>;

G:=Group( (1,4)(2,3)(5,7)(6,8), (1,8)(2,5)(3,7)(4,6), (1,8)(4,6), (1,6)(4,8), (3,5,7)(4,8,6), (1,2)(3,4,5,6,7,8) );

G=PermutationGroup([(1,4),(2,3),(5,7),(6,8)], [(1,8),(2,5),(3,7),(4,6)], [(1,8),(4,6)], [(1,6),(4,8)], [(3,5,7),(4,8,6)], [(1,2),(3,4,5,6,7,8)])

G:=TransitiveGroup(8,42);

On 12 points - transitive group 12T126
Generators in S12
(1 10)(2 11)(3 12)(6 9)
(1 10)(2 11)(4 7)(5 8)
(1 10)(5 8)
(3 12)(5 8)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)

G:=sub<Sym(12)| (1,10)(2,11)(3,12)(6,9), (1,10)(2,11)(4,7)(5,8), (1,10)(5,8), (3,12)(5,8), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12)>;

G:=Group( (1,10)(2,11)(3,12)(6,9), (1,10)(2,11)(4,7)(5,8), (1,10)(5,8), (3,12)(5,8), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12) );

G=PermutationGroup([(1,10),(2,11),(3,12),(6,9)], [(1,10),(2,11),(4,7),(5,8)], [(1,10),(5,8)], [(3,12),(5,8)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)])

G:=TransitiveGroup(12,126);

On 12 points - transitive group 12T128
Generators in S12
(1 4)(2 12)(5 6)(8 10)
(2 12)(3 9)(7 11)(8 10)
(1 4)(2 8)(3 7)(5 6)(9 11)(10 12)
(1 5)(2 12)(3 11)(4 6)(7 9)(8 10)
(1 2 3)(4 12 9)(5 10 7)(6 8 11)
(2 3)(4 5 6)(7 8 9 10 11 12)

G:=sub<Sym(12)| (1,4)(2,12)(5,6)(8,10), (2,12)(3,9)(7,11)(8,10), (1,4)(2,8)(3,7)(5,6)(9,11)(10,12), (1,5)(2,12)(3,11)(4,6)(7,9)(8,10), (1,2,3)(4,12,9)(5,10,7)(6,8,11), (2,3)(4,5,6)(7,8,9,10,11,12)>;

G:=Group( (1,4)(2,12)(5,6)(8,10), (2,12)(3,9)(7,11)(8,10), (1,4)(2,8)(3,7)(5,6)(9,11)(10,12), (1,5)(2,12)(3,11)(4,6)(7,9)(8,10), (1,2,3)(4,12,9)(5,10,7)(6,8,11), (2,3)(4,5,6)(7,8,9,10,11,12) );

G=PermutationGroup([(1,4),(2,12),(5,6),(8,10)], [(2,12),(3,9),(7,11),(8,10)], [(1,4),(2,8),(3,7),(5,6),(9,11),(10,12)], [(1,5),(2,12),(3,11),(4,6),(7,9),(8,10)], [(1,2,3),(4,12,9),(5,10,7),(6,8,11)], [(2,3),(4,5,6),(7,8,9,10,11,12)])

G:=TransitiveGroup(12,128);

On 12 points - transitive group 12T129
Generators in S12
(2 7)(3 11)(5 10)(6 8)
(2 5)(3 6)(7 10)(8 11)
(1 4)(2 7)(3 8)(5 10)(6 11)(9 12)
(1 12)(2 5)(3 11)(4 9)(6 8)(7 10)
(4 12 9)(5 10 7)(6 8 11)
(1 2 3)(4 5 6)(7 8 9 10 11 12)

G:=sub<Sym(12)| (2,7)(3,11)(5,10)(6,8), (2,5)(3,6)(7,10)(8,11), (1,4)(2,7)(3,8)(5,10)(6,11)(9,12), (1,12)(2,5)(3,11)(4,9)(6,8)(7,10), (4,12,9)(5,10,7)(6,8,11), (1,2,3)(4,5,6)(7,8,9,10,11,12)>;

G:=Group( (2,7)(3,11)(5,10)(6,8), (2,5)(3,6)(7,10)(8,11), (1,4)(2,7)(3,8)(5,10)(6,11)(9,12), (1,12)(2,5)(3,11)(4,9)(6,8)(7,10), (4,12,9)(5,10,7)(6,8,11), (1,2,3)(4,5,6)(7,8,9,10,11,12) );

G=PermutationGroup([(2,7),(3,11),(5,10),(6,8)], [(2,5),(3,6),(7,10),(8,11)], [(1,4),(2,7),(3,8),(5,10),(6,11),(9,12)], [(1,12),(2,5),(3,11),(4,9),(6,8),(7,10)], [(4,12,9),(5,10,7),(6,8,11)], [(1,2,3),(4,5,6),(7,8,9,10,11,12)])

G:=TransitiveGroup(12,129);

On 16 points: primitive - transitive group 16T708
Generators in S16
(1 7)(2 10)(3 14)(4 11)(5 9)(6 12)(8 13)(15 16)
(1 2)(3 4)(5 15)(6 13)(7 10)(8 12)(9 16)(11 14)
(1 14)(2 11)(3 7)(4 10)(5 13)(6 15)(8 9)(12 16)
(1 12)(2 8)(3 15)(4 5)(6 7)(9 11)(10 13)(14 16)
(2 10 7)(3 8 5)(4 6 9)(11 13 15)(12 16 14)
(2 3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)

G:=sub<Sym(16)| (1,7)(2,10)(3,14)(4,11)(5,9)(6,12)(8,13)(15,16), (1,2)(3,4)(5,15)(6,13)(7,10)(8,12)(9,16)(11,14), (1,14)(2,11)(3,7)(4,10)(5,13)(6,15)(8,9)(12,16), (1,12)(2,8)(3,15)(4,5)(6,7)(9,11)(10,13)(14,16), (2,10,7)(3,8,5)(4,6,9)(11,13,15)(12,16,14), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16)>;

G:=Group( (1,7)(2,10)(3,14)(4,11)(5,9)(6,12)(8,13)(15,16), (1,2)(3,4)(5,15)(6,13)(7,10)(8,12)(9,16)(11,14), (1,14)(2,11)(3,7)(4,10)(5,13)(6,15)(8,9)(12,16), (1,12)(2,8)(3,15)(4,5)(6,7)(9,11)(10,13)(14,16), (2,10,7)(3,8,5)(4,6,9)(11,13,15)(12,16,14), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16) );

G=PermutationGroup([(1,7),(2,10),(3,14),(4,11),(5,9),(6,12),(8,13),(15,16)], [(1,2),(3,4),(5,15),(6,13),(7,10),(8,12),(9,16),(11,14)], [(1,14),(2,11),(3,7),(4,10),(5,13),(6,15),(8,9),(12,16)], [(1,12),(2,8),(3,15),(4,5),(6,7),(9,11),(10,13),(14,16)], [(2,10,7),(3,8,5),(4,6,9),(11,13,15),(12,16,14)], [(2,3,4),(5,6,7,8,9,10),(11,12,13,14,15,16)])

G:=TransitiveGroup(16,708);

On 18 points - transitive group 18T112
Generators in S18
(1 4)(2 5)(10 17)(12 13)
(7 14)(9 16)(10 17)(12 13)
(1 4)(2 5)(7 14)(8 15)(11 18)(12 13)
(2 5)(3 6)(9 16)(10 17)(11 18)(12 13)
(1 13 9)(2 10 14)(3 15 11)(4 12 16)(5 17 7)(6 8 18)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

G:=sub<Sym(18)| (1,4)(2,5)(10,17)(12,13), (7,14)(9,16)(10,17)(12,13), (1,4)(2,5)(7,14)(8,15)(11,18)(12,13), (2,5)(3,6)(9,16)(10,17)(11,18)(12,13), (1,13,9)(2,10,14)(3,15,11)(4,12,16)(5,17,7)(6,8,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;

G:=Group( (1,4)(2,5)(10,17)(12,13), (7,14)(9,16)(10,17)(12,13), (1,4)(2,5)(7,14)(8,15)(11,18)(12,13), (2,5)(3,6)(9,16)(10,17)(11,18)(12,13), (1,13,9)(2,10,14)(3,15,11)(4,12,16)(5,17,7)(6,8,18), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );

G=PermutationGroup([(1,4),(2,5),(10,17),(12,13)], [(7,14),(9,16),(10,17),(12,13)], [(1,4),(2,5),(7,14),(8,15),(11,18),(12,13)], [(2,5),(3,6),(9,16),(10,17),(11,18),(12,13)], [(1,13,9),(2,10,14),(3,15,11),(4,12,16),(5,17,7),(6,8,18)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)])

G:=TransitiveGroup(18,112);

On 18 points - transitive group 18T113
Generators in S18
(1 5)(2 6)(8 15)(12 13)
(8 15)(9 16)(11 18)(12 13)
(1 5)(2 6)(7 14)(8 15)(9 16)(10 17)
(2 6)(3 4)(7 14)(8 15)(11 18)(12 13)
(1 15 18)(2 13 16)(3 17 14)(4 10 7)(5 8 11)(6 12 9)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

G:=sub<Sym(18)| (1,5)(2,6)(8,15)(12,13), (8,15)(9,16)(11,18)(12,13), (1,5)(2,6)(7,14)(8,15)(9,16)(10,17), (2,6)(3,4)(7,14)(8,15)(11,18)(12,13), (1,15,18)(2,13,16)(3,17,14)(4,10,7)(5,8,11)(6,12,9), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;

G:=Group( (1,5)(2,6)(8,15)(12,13), (8,15)(9,16)(11,18)(12,13), (1,5)(2,6)(7,14)(8,15)(9,16)(10,17), (2,6)(3,4)(7,14)(8,15)(11,18)(12,13), (1,15,18)(2,13,16)(3,17,14)(4,10,7)(5,8,11)(6,12,9), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );

G=PermutationGroup([(1,5),(2,6),(8,15),(12,13)], [(8,15),(9,16),(11,18),(12,13)], [(1,5),(2,6),(7,14),(8,15),(9,16),(10,17)], [(2,6),(3,4),(7,14),(8,15),(11,18),(12,13)], [(1,15,18),(2,13,16),(3,17,14),(4,10,7),(5,8,11),(6,12,9)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)])

G:=TransitiveGroup(18,113);

On 24 points - transitive group 24T692
Generators in S24
(1 10)(2 21)(3 22)(4 15)(5 16)(6 9)(7 23)(8 24)(11 13)(12 14)(17 19)(18 20)
(1 20)(2 11)(3 14)(4 23)(5 8)(6 17)(7 15)(9 19)(10 18)(12 22)(13 21)(16 24)
(1 20)(3 14)(5 8)(10 18)(12 22)(16 24)
(1 18)(3 12)(5 24)(8 16)(10 20)(14 22)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)(13 15 17)(14 18 16)(19 21 23)(20 24 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)| (1,10)(2,21)(3,22)(4,15)(5,16)(6,9)(7,23)(8,24)(11,13)(12,14)(17,19)(18,20), (1,20)(2,11)(3,14)(4,23)(5,8)(6,17)(7,15)(9,19)(10,18)(12,22)(13,21)(16,24), (1,20)(3,14)(5,8)(10,18)(12,22)(16,24), (1,18)(3,12)(5,24)(8,16)(10,20)(14,22), (1,5,3)(2,4,6)(7,9,11)(8,12,10)(13,15,17)(14,18,16)(19,21,23)(20,24,22), (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( (1,10)(2,21)(3,22)(4,15)(5,16)(6,9)(7,23)(8,24)(11,13)(12,14)(17,19)(18,20), (1,20)(2,11)(3,14)(4,23)(5,8)(6,17)(7,15)(9,19)(10,18)(12,22)(13,21)(16,24), (1,20)(3,14)(5,8)(10,18)(12,22)(16,24), (1,18)(3,12)(5,24)(8,16)(10,20)(14,22), (1,5,3)(2,4,6)(7,9,11)(8,12,10)(13,15,17)(14,18,16)(19,21,23)(20,24,22), (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([(1,10),(2,21),(3,22),(4,15),(5,16),(6,9),(7,23),(8,24),(11,13),(12,14),(17,19),(18,20)], [(1,20),(2,11),(3,14),(4,23),(5,8),(6,17),(7,15),(9,19),(10,18),(12,22),(13,21),(16,24)], [(1,20),(3,14),(5,8),(10,18),(12,22),(16,24)], [(1,18),(3,12),(5,24),(8,16),(10,20),(14,22)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10),(13,15,17),(14,18,16),(19,21,23),(20,24,22)], [(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(24,692);

On 24 points - transitive group 24T694
Generators in S24
(1 15)(2 18)(3 20)(5 10)(8 12)(13 17)(14 16)(22 24)
(3 20)(4 23)(5 10)(6 7)(8 12)(9 11)(19 21)(22 24)
(1 15)(2 18)(3 22)(4 21)(5 12)(6 11)(7 9)(8 10)(13 17)(14 16)(19 23)(20 24)
(1 13)(2 16)(3 20)(4 19)(5 10)(6 9)(7 11)(8 12)(14 18)(15 17)(21 23)(22 24)
(1 5 4)(2 3 6)(7 18 20)(8 21 13)(9 14 22)(10 23 15)(11 16 24)(12 19 17)
(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)(2,18)(3,20)(5,10)(8,12)(13,17)(14,16)(22,24), (3,20)(4,23)(5,10)(6,7)(8,12)(9,11)(19,21)(22,24), (1,15)(2,18)(3,22)(4,21)(5,12)(6,11)(7,9)(8,10)(13,17)(14,16)(19,23)(20,24), (1,13)(2,16)(3,20)(4,19)(5,10)(6,9)(7,11)(8,12)(14,18)(15,17)(21,23)(22,24), (1,5,4)(2,3,6)(7,18,20)(8,21,13)(9,14,22)(10,23,15)(11,16,24)(12,19,17), (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)(2,18)(3,20)(5,10)(8,12)(13,17)(14,16)(22,24), (3,20)(4,23)(5,10)(6,7)(8,12)(9,11)(19,21)(22,24), (1,15)(2,18)(3,22)(4,21)(5,12)(6,11)(7,9)(8,10)(13,17)(14,16)(19,23)(20,24), (1,13)(2,16)(3,20)(4,19)(5,10)(6,9)(7,11)(8,12)(14,18)(15,17)(21,23)(22,24), (1,5,4)(2,3,6)(7,18,20)(8,21,13)(9,14,22)(10,23,15)(11,16,24)(12,19,17), (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),(2,18),(3,20),(5,10),(8,12),(13,17),(14,16),(22,24)], [(3,20),(4,23),(5,10),(6,7),(8,12),(9,11),(19,21),(22,24)], [(1,15),(2,18),(3,22),(4,21),(5,12),(6,11),(7,9),(8,10),(13,17),(14,16),(19,23),(20,24)], [(1,13),(2,16),(3,20),(4,19),(5,10),(6,9),(7,11),(8,12),(14,18),(15,17),(21,23),(22,24)], [(1,5,4),(2,3,6),(7,18,20),(8,21,13),(9,14,22),(10,23,15),(11,16,24),(12,19,17)], [(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(24,694);

On 24 points - transitive group 24T695
Generators in S24
(1 20)(2 13)(4 15)(5 24)(8 18)(9 21)(11 23)(12 16)
(1 8)(2 9)(4 11)(5 12)(13 21)(15 23)(16 24)(18 20)
(1 20)(2 21)(3 10)(4 15)(5 16)(6 7)(8 18)(9 13)(11 23)(12 24)(14 22)(17 19)
(1 8)(2 13)(3 14)(4 11)(5 24)(6 19)(7 17)(9 21)(10 22)(12 16)(15 23)(18 20)
(7 19 17)(8 18 20)(9 21 13)(10 14 22)(11 23 15)(12 16 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)| (1,20)(2,13)(4,15)(5,24)(8,18)(9,21)(11,23)(12,16), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(18,20), (1,20)(2,21)(3,10)(4,15)(5,16)(6,7)(8,18)(9,13)(11,23)(12,24)(14,22)(17,19), (1,8)(2,13)(3,14)(4,11)(5,24)(6,19)(7,17)(9,21)(10,22)(12,16)(15,23)(18,20), (7,19,17)(8,18,20)(9,21,13)(10,14,22)(11,23,15)(12,16,24), (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( (1,20)(2,13)(4,15)(5,24)(8,18)(9,21)(11,23)(12,16), (1,8)(2,9)(4,11)(5,12)(13,21)(15,23)(16,24)(18,20), (1,20)(2,21)(3,10)(4,15)(5,16)(6,7)(8,18)(9,13)(11,23)(12,24)(14,22)(17,19), (1,8)(2,13)(3,14)(4,11)(5,24)(6,19)(7,17)(9,21)(10,22)(12,16)(15,23)(18,20), (7,19,17)(8,18,20)(9,21,13)(10,14,22)(11,23,15)(12,16,24), (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([(1,20),(2,13),(4,15),(5,24),(8,18),(9,21),(11,23),(12,16)], [(1,8),(2,9),(4,11),(5,12),(13,21),(15,23),(16,24),(18,20)], [(1,20),(2,21),(3,10),(4,15),(5,16),(6,7),(8,18),(9,13),(11,23),(12,24),(14,22),(17,19)], [(1,8),(2,13),(3,14),(4,11),(5,24),(6,19),(7,17),(9,21),(10,22),(12,16),(15,23),(18,20)], [(7,19,17),(8,18,20),(9,21,13),(10,14,22),(11,23,15),(12,16,24)], [(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(24,695);

On 24 points - transitive group 24T702
Generators in S24
(1 20)(2 21)(3 15)(4 16)(5 10)(6 11)(7 14)(8 22)(9 23)(12 13)(17 24)(18 19)
(1 12)(2 14)(3 22)(4 9)(5 17)(6 19)(7 21)(8 15)(10 24)(11 18)(13 20)(16 23)
(2 14)(4 9)(6 19)(7 21)(11 18)(16 23)
(2 7)(4 23)(6 18)(9 16)(11 19)(14 21)
(1 5 3)(2 4 6)(7 16 19)(8 20 17)(9 18 21)(10 22 13)(11 14 23)(12 24 15)
(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)| (1,20)(2,21)(3,15)(4,16)(5,10)(6,11)(7,14)(8,22)(9,23)(12,13)(17,24)(18,19), (1,12)(2,14)(3,22)(4,9)(5,17)(6,19)(7,21)(8,15)(10,24)(11,18)(13,20)(16,23), (2,14)(4,9)(6,19)(7,21)(11,18)(16,23), (2,7)(4,23)(6,18)(9,16)(11,19)(14,21), (1,5,3)(2,4,6)(7,16,19)(8,20,17)(9,18,21)(10,22,13)(11,14,23)(12,24,15), (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( (1,20)(2,21)(3,15)(4,16)(5,10)(6,11)(7,14)(8,22)(9,23)(12,13)(17,24)(18,19), (1,12)(2,14)(3,22)(4,9)(5,17)(6,19)(7,21)(8,15)(10,24)(11,18)(13,20)(16,23), (2,14)(4,9)(6,19)(7,21)(11,18)(16,23), (2,7)(4,23)(6,18)(9,16)(11,19)(14,21), (1,5,3)(2,4,6)(7,16,19)(8,20,17)(9,18,21)(10,22,13)(11,14,23)(12,24,15), (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([(1,20),(2,21),(3,15),(4,16),(5,10),(6,11),(7,14),(8,22),(9,23),(12,13),(17,24),(18,19)], [(1,12),(2,14),(3,22),(4,9),(5,17),(6,19),(7,21),(8,15),(10,24),(11,18),(13,20),(16,23)], [(2,14),(4,9),(6,19),(7,21),(11,18),(16,23)], [(2,7),(4,23),(6,18),(9,16),(11,19),(14,21)], [(1,5,3),(2,4,6),(7,16,19),(8,20,17),(9,18,21),(10,22,13),(11,14,23),(12,24,15)], [(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(24,702);

On 24 points - transitive group 24T703
Generators in S24
(1 24)(2 19)(3 10)(4 11)(5 16)(6 17)(7 23)(8 18)(9 13)(12 22)(14 20)(15 21)
(1 18)(2 9)(3 20)(4 15)(5 12)(6 23)(7 17)(8 24)(10 14)(11 21)(13 19)(16 22)
(2 9)(4 15)(6 23)(7 17)(11 21)(13 19)
(2 13)(4 21)(6 7)(9 19)(11 15)(17 23)
(7 17 23)(8 24 18)(9 13 19)(10 20 14)(11 15 21)(12 22 16)
(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)| (1,24)(2,19)(3,10)(4,11)(5,16)(6,17)(7,23)(8,18)(9,13)(12,22)(14,20)(15,21), (1,18)(2,9)(3,20)(4,15)(5,12)(6,23)(7,17)(8,24)(10,14)(11,21)(13,19)(16,22), (2,9)(4,15)(6,23)(7,17)(11,21)(13,19), (2,13)(4,21)(6,7)(9,19)(11,15)(17,23), (7,17,23)(8,24,18)(9,13,19)(10,20,14)(11,15,21)(12,22,16), (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( (1,24)(2,19)(3,10)(4,11)(5,16)(6,17)(7,23)(8,18)(9,13)(12,22)(14,20)(15,21), (1,18)(2,9)(3,20)(4,15)(5,12)(6,23)(7,17)(8,24)(10,14)(11,21)(13,19)(16,22), (2,9)(4,15)(6,23)(7,17)(11,21)(13,19), (2,13)(4,21)(6,7)(9,19)(11,15)(17,23), (7,17,23)(8,24,18)(9,13,19)(10,20,14)(11,15,21)(12,22,16), (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([(1,24),(2,19),(3,10),(4,11),(5,16),(6,17),(7,23),(8,18),(9,13),(12,22),(14,20),(15,21)], [(1,18),(2,9),(3,20),(4,15),(5,12),(6,23),(7,17),(8,24),(10,14),(11,21),(13,19),(16,22)], [(2,9),(4,15),(6,23),(7,17),(11,21),(13,19)], [(2,13),(4,21),(6,7),(9,19),(11,15),(17,23)], [(7,17,23),(8,24,18),(9,13,19),(10,20,14),(11,15,21),(12,22,16)], [(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(24,703);

On 24 points - transitive group 24T704
Generators in S24
(1 24)(2 19)(3 9)(4 10)(5 17)(6 18)(7 11)(8 12)(13 15)(14 16)(20 22)(21 23)
(1 20)(2 23)(3 11)(4 8)(5 13)(6 16)(7 9)(10 12)(14 18)(15 17)(19 21)(22 24)
(2 23)(4 8)(6 16)(10 12)(14 18)(19 21)
(2 21)(4 12)(6 14)(8 10)(16 18)(19 23)
(1 5 3)(2 4 6)(7 24 13)(8 14 19)(9 20 15)(10 16 21)(11 22 17)(12 18 23)
(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)| (1,24)(2,19)(3,9)(4,10)(5,17)(6,18)(7,11)(8,12)(13,15)(14,16)(20,22)(21,23), (1,20)(2,23)(3,11)(4,8)(5,13)(6,16)(7,9)(10,12)(14,18)(15,17)(19,21)(22,24), (2,23)(4,8)(6,16)(10,12)(14,18)(19,21), (2,21)(4,12)(6,14)(8,10)(16,18)(19,23), (1,5,3)(2,4,6)(7,24,13)(8,14,19)(9,20,15)(10,16,21)(11,22,17)(12,18,23), (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( (1,24)(2,19)(3,9)(4,10)(5,17)(6,18)(7,11)(8,12)(13,15)(14,16)(20,22)(21,23), (1,20)(2,23)(3,11)(4,8)(5,13)(6,16)(7,9)(10,12)(14,18)(15,17)(19,21)(22,24), (2,23)(4,8)(6,16)(10,12)(14,18)(19,21), (2,21)(4,12)(6,14)(8,10)(16,18)(19,23), (1,5,3)(2,4,6)(7,24,13)(8,14,19)(9,20,15)(10,16,21)(11,22,17)(12,18,23), (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([(1,24),(2,19),(3,9),(4,10),(5,17),(6,18),(7,11),(8,12),(13,15),(14,16),(20,22),(21,23)], [(1,20),(2,23),(3,11),(4,8),(5,13),(6,16),(7,9),(10,12),(14,18),(15,17),(19,21),(22,24)], [(2,23),(4,8),(6,16),(10,12),(14,18),(19,21)], [(2,21),(4,12),(6,14),(8,10),(16,18),(19,23)], [(1,5,3),(2,4,6),(7,24,13),(8,14,19),(9,20,15),(10,16,21),(11,22,17),(12,18,23)], [(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(24,704);

Polynomial with Galois group A4≀C2 over ℚ
actionf(x)Disc(f)
8T42x8-2x7-12x6+16x5+52x4-26x3-80x2-24x-1218·54·3792
12T126x12-12x10+48x8-80x6+54x4-13x2+1212·56·1394
12T128x12-64x10-24x9+1450x8+1024x7-13572x6-13376x5+45009x4+58784x3-8716x2-28424x-6776260·34·72·112·1038·14812·154512·241132
12T129x12-62x10-48x9+1114x8+1984x7-5232x6-14872x5-4455x4+12320x3+8780x2+1380x-73227·32·1038·2292·38772·40032·22587912

Matrix representation of A4≀C2 in GL6(ℤ)

001000
-1-1-1000
100000
000010
000100
000-1-1-1
,
-1-1-1000
001000
010000
000-1-1-1
000001
000010
,
100000
010000
001000
000-1-1-1
000001
000010
,
100000
010000
001000
000001
000-1-1-1
000100
,
100000
-1-1-1000
010000
000100
000001
000-1-1-1
,
000100
000001
000-1-1-1
100000
001000
-1-1-1000

G:=sub<GL(6,Integers())| [0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,1,0,0,0,-1,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0],[1,-1,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1],[0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1,1,0,-1,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0] >;

A4≀C2 in GAP, Magma, Sage, TeX

A_4\wr C_2
% in TeX

G:=Group("A4wrC2");
// GroupNames label

G:=SmallGroup(288,1025);
// by ID

G=gap.SmallGroup(288,1025);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,2,2,254,2019,766,185,5044,326,333,761,4548,1531,1770,1777,608]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^3=f^6=1,e*a*e^-1=a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,f*d*f^-1=b*c=c*b,b*d=d*b,e*b*e^-1=a,f*b*f^-1=a*b*c,e*c*e^-1=c*d=d*c,f*c*f^-1=a*b*d,e*d*e^-1=c,f*e*f^-1=e^-1>;
// generators/relations

Export

Subgroup lattice of A4≀C2 in TeX
Character table of A4≀C2 in TeX

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