A4wrC2 - GroupNames

G = A₄wrC₂ order 288 = 2⁵·3²

Wreath product of A₄ by C₂

non-abelian, soluble, monomial

Aliases: A₄ wr C₂, A₄²:₁C₂, C₂²:A₄:C₆, C₂²:S₄:C₃, C₂²:A₄:₁S₃, C₂⁴:₂(C₃xS₃), SmallGroup(288,1025)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²:A₄ — A₄wrC₂

Chief series

C₁ — C₂⁴ — C₂²:A₄ — A₄² — A₄wrC₂

Lower central

C₂²:A₄ — A₄wrC₂

Upper central

C₁

Generators and relations for A₄wrC₂
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 >

Subgroups: 442 in 46 conjugacy classes, 7 normal (all characteristic)
Quotients: C₁, C₂, C₃, S₃, C₆, C₃xS₃, A₄wrC₂

C₁

₆C₂

₉C₂

₁₂C₂

₈C₃

₁₆C₃

₂C₂²

₃C₂²

₉C₂²

₁₈C₂²

₁₈C₄

₁₆S₃

₂₄C₆

₄₈C₆

₁₆C₃²

₃C₂³

₆C₂³

₉C₂xC₄

₉C₂³

₁₈D₄

₂A₄

₈C₂xC₆

₈A₄

₁₂A₄

₁₆C₃xS₃

C₂⁴

₉C₂xD₄

₉C₂²:C₄

₆C₂xA₄

₁₂C₂xA₄

₁₂S₄

₈C₃xA₄

₃C₂²wrC₂

C₂²:A₄

₂C₂²xA₄

C₂²:S₄

₃C₂⁴:C₆

A₄²

A₄wrC₂

Character table of A₄wrC₂

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4	6A	6B	6C	6D
size	1	6	9	12	8	8	16	16	32	36	24	24	48	48
ρ₁	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	linear of order 2
ρ₃	1	1	1	-1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	-1	ζ₃	ζ₃²	ζ₆⁵	ζ₆	linear of order 6
ρ₄	1	1	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₅	1	1	1	-1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	-1	ζ₃²	ζ₃	ζ₆	ζ₆⁵	linear of order 6
ρ₆	1	1	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₇	2	2	2	0	-1	-1	2	2	-1	0	-1	-1	0	0	orthogonal lifted from S₃
ρ₈	2	2	2	0	ζ₆⁵	ζ₆	-1+√-3	-1-√-3	-1	0	ζ₆	ζ₆⁵	0	0	complex lifted from C₃xS₃
ρ₉	2	2	2	0	ζ₆	ζ₆⁵	-1-√-3	-1+√-3	-1	0	ζ₆⁵	ζ₆	0	0	complex lifted from C₃xS₃
ρ₁₀	6	2	-2	0	3	3	0	0	0	0	-1	-1	0	0	orthogonal faithful
ρ₁₁	6	2	-2	0	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	ζ₆⁵	ζ₆	0	0	complex faithful
ρ₁₂	6	2	-2	0	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	ζ₆	ζ₆⁵	0	0	complex faithful
ρ₁₃	9	-3	1	3	0	0	0	0	0	-1	0	0	0	0	orthogonal faithful
ρ₁₄	9	-3	1	-3	0	0	0	0	0	1	0	0	0	0	orthogonal faithful

Permutation representations of A₄wrC₂
►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 7)(5 6)(9 11)

(2 7)(3 10)(8 12)(9 11)

(1 4)(2 9)(3 8)(5 6)(7 11)(10 12)

(1 5)(2 7)(3 12)(4 6)(8 10)(9 11)

(1 2 3)(4 7 10)(5 11 8)(6 9 12)

(2 3)(4 5 6)(7 8 9 10 11 12)

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

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

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

G:=TransitiveGroup(12,128);

►On 12 points - transitive group 12T129

Generators in S₁₂

(1 7)(3 9)(5 12)(6 10)

(1 6)(3 5)(7 10)(9 12)

(1 10)(2 4)(3 9)(5 12)(6 7)(8 11)

(1 7)(2 8)(3 5)(4 11)(6 10)(9 12)

(4 8 11)(5 12 9)(6 10 7)

(1 2 3)(4 5 6)(7 8 9 10 11 12)

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

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

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

(1 2)(3 4)(5 16)(6 9)(7 15)(8 13)(10 12)(11 14)

(1 14)(2 11)(3 9)(4 6)(5 10)(7 13)(8 15)(12 16)

(1 12)(2 10)(3 15)(4 7)(5 11)(6 13)(8 9)(14 16)

(2 6 9)(3 10 7)(4 8 5)(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)| (1,9)(2,6)(3,14)(4,11)(5,7)(8,12)(10,13)(15,16), (1,2)(3,4)(5,16)(6,9)(7,15)(8,13)(10,12)(11,14), (1,14)(2,11)(3,9)(4,6)(5,10)(7,13)(8,15)(12,16), (1,12)(2,10)(3,15)(4,7)(5,11)(6,13)(8,9)(14,16), (2,6,9)(3,10,7)(4,8,5)(11,13,15)(12,16,14), (2,3,4)(5,6,7,8,9,10)(11,12,13,14,15,16)>;

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

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

(7 13)(9 15)(10 16)(12 18)

(1 4)(2 5)(7 13)(8 14)(11 17)(12 18)

(2 5)(3 6)(9 15)(10 16)(11 17)(12 18)

(1 18 9)(2 10 13)(3 14 11)(4 12 15)(5 16 7)(6 8 17)

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

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

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

(7 15)(9 17)(10 18)(12 14)

(1 5)(2 6)(8 16)(9 17)(10 18)(11 13)

(2 6)(3 4)(7 15)(8 16)(9 17)(12 14)

(1 17 14)(2 15 18)(3 13 16)(4 11 8)(5 9 12)(6 7 10)

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

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

G=PermutationGroup([[(1,5),(2,6),(7,15),(9,17)], [(7,15),(9,17),(10,18),(12,14)], [(1,5),(2,6),(8,16),(9,17),(10,18),(11,13)], [(2,6),(3,4),(7,15),(8,16),(9,17),(12,14)], [(1,17,14),(2,15,18),(3,13,16),(4,11,8),(5,9,12),(6,7,10)], [(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 14)(2 21)(3 22)(4 9)(5 10)(6 13)(7 15)(8 16)(11 19)(12 20)(17 23)(18 24)

(1 20)(2 15)(3 8)(4 23)(5 18)(6 11)(7 21)(9 17)(10 24)(12 14)(13 19)(16 22)

(1 20)(3 8)(5 18)(10 24)(12 14)(16 22)

(1 12)(3 16)(5 24)(8 22)(10 18)(14 20)

(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)| (1,14)(2,21)(3,22)(4,9)(5,10)(6,13)(7,15)(8,16)(11,19)(12,20)(17,23)(18,24), (1,20)(2,15)(3,8)(4,23)(5,18)(6,11)(7,21)(9,17)(10,24)(12,14)(13,19)(16,22), (1,20)(3,8)(5,18)(10,24)(12,14)(16,22), (1,12)(3,16)(5,24)(8,22)(10,18)(14,20), (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( (1,14)(2,21)(3,22)(4,9)(5,10)(6,13)(7,15)(8,16)(11,19)(12,20)(17,23)(18,24), (1,20)(2,15)(3,8)(4,23)(5,18)(6,11)(7,21)(9,17)(10,24)(12,14)(13,19)(16,22), (1,20)(3,8)(5,18)(10,24)(12,14)(16,22), (1,12)(3,16)(5,24)(8,22)(10,18)(14,20), (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([[(1,14),(2,21),(3,22),(4,9),(5,10),(6,13),(7,15),(8,16),(11,19),(12,20),(17,23),(18,24)], [(1,20),(2,15),(3,8),(4,23),(5,18),(6,11),(7,21),(9,17),(10,24),(12,14),(13,19),(16,22)], [(1,20),(3,8),(5,18),(10,24),(12,14),(16,22)], [(1,12),(3,16),(5,24),(8,22),(10,18),(14,20)], [(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 18)(2 15)(3 21)(5 9)(7 11)(13 17)(14 16)(19 23)

(3 21)(4 24)(5 9)(6 12)(7 11)(8 10)(19 23)(20 22)

(1 18)(2 15)(3 23)(4 22)(5 11)(6 10)(7 9)(8 12)(13 17)(14 16)(19 21)(20 24)

(1 16)(2 13)(3 21)(4 20)(5 9)(6 8)(7 11)(10 12)(14 18)(15 17)(19 23)(22 24)

(1 5 4)(2 3 6)(7 22 16)(8 17 23)(9 24 18)(10 13 19)(11 20 14)(12 15 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)| (1,18)(2,15)(3,21)(5,9)(7,11)(13,17)(14,16)(19,23), (3,21)(4,24)(5,9)(6,12)(7,11)(8,10)(19,23)(20,22), (1,18)(2,15)(3,23)(4,22)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(19,21)(20,24), (1,16)(2,13)(3,21)(4,20)(5,9)(6,8)(7,11)(10,12)(14,18)(15,17)(19,23)(22,24), (1,5,4)(2,3,6)(7,22,16)(8,17,23)(9,24,18)(10,13,19)(11,20,14)(12,15,21), (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( (1,18)(2,15)(3,21)(5,9)(7,11)(13,17)(14,16)(19,23), (3,21)(4,24)(5,9)(6,12)(7,11)(8,10)(19,23)(20,22), (1,18)(2,15)(3,23)(4,22)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(19,21)(20,24), (1,16)(2,13)(3,21)(4,20)(5,9)(6,8)(7,11)(10,12)(14,18)(15,17)(19,23)(22,24), (1,5,4)(2,3,6)(7,22,16)(8,17,23)(9,24,18)(10,13,19)(11,20,14)(12,15,21), (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([[(1,18),(2,15),(3,21),(5,9),(7,11),(13,17),(14,16),(19,23)], [(3,21),(4,24),(5,9),(6,12),(7,11),(8,10),(19,23),(20,22)], [(1,18),(2,15),(3,23),(4,22),(5,11),(6,10),(7,9),(8,12),(13,17),(14,16),(19,21),(20,24)], [(1,16),(2,13),(3,21),(4,20),(5,9),(6,8),(7,11),(10,12),(14,18),(15,17),(19,23),(22,24)], [(1,5,4),(2,3,6),(7,22,16),(8,17,23),(9,24,18),(10,13,19),(11,20,14),(12,15,21)], [(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(24,694);

►On 24 points - transitive group 24T695

Generators in S₂₄

(1 8)(2 13)(4 15)(5 12)(9 21)(11 23)(16 24)(18 20)

(1 20)(2 21)(4 23)(5 24)(8 18)(9 13)(11 15)(12 16)

(1 8)(2 9)(3 22)(4 15)(5 16)(6 19)(7 17)(10 14)(11 23)(12 24)(13 21)(18 20)

(1 20)(2 13)(3 14)(4 23)(5 12)(6 7)(8 18)(9 21)(10 22)(11 15)(16 24)(17 19)

(7 17 19)(8 20 18)(9 13 21)(10 22 14)(11 15 23)(12 24 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)| (1,8)(2,13)(4,15)(5,12)(9,21)(11,23)(16,24)(18,20), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (7,17,19)(8,20,18)(9,13,21)(10,22,14)(11,15,23)(12,24,16), (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( (1,8)(2,13)(4,15)(5,12)(9,21)(11,23)(16,24)(18,20), (1,20)(2,21)(4,23)(5,24)(8,18)(9,13)(11,15)(12,16), (1,8)(2,9)(3,22)(4,15)(5,16)(6,19)(7,17)(10,14)(11,23)(12,24)(13,21)(18,20), (1,20)(2,13)(3,14)(4,23)(5,12)(6,7)(8,18)(9,21)(10,22)(11,15)(16,24)(17,19), (7,17,19)(8,20,18)(9,13,21)(10,22,14)(11,15,23)(12,24,16), (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([[(1,8),(2,13),(4,15),(5,12),(9,21),(11,23),(16,24),(18,20)], [(1,20),(2,21),(4,23),(5,24),(8,18),(9,13),(11,15),(12,16)], [(1,8),(2,9),(3,22),(4,15),(5,16),(6,19),(7,17),(10,14),(11,23),(12,24),(13,21),(18,20)], [(1,20),(2,13),(3,14),(4,23),(5,12),(6,7),(8,18),(9,21),(10,22),(11,15),(16,24),(17,19)], [(7,17,19),(8,20,18),(9,13,21),(10,22,14),(11,15,23),(12,24,16)], [(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(24,695);

►On 24 points - transitive group 24T702

Generators in S₂₄

(1 20)(2 21)(3 18)(4 13)(5 10)(6 11)(7 17)(8 22)(9 23)(12 16)(14 24)(15 19)

(1 12)(2 17)(3 22)(4 9)(5 14)(6 19)(7 21)(8 18)(10 24)(11 15)(13 23)(16 20)

(2 17)(4 9)(6 19)(7 21)(11 15)(13 23)

(2 7)(4 23)(6 15)(9 13)(11 19)(17 21)

(1 5 3)(2 4 6)(7 13 19)(8 20 14)(9 15 21)(10 22 16)(11 17 23)(12 24 18)

(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)(3,18)(4,13)(5,10)(6,11)(7,17)(8,22)(9,23)(12,16)(14,24)(15,19), (1,12)(2,17)(3,22)(4,9)(5,14)(6,19)(7,21)(8,18)(10,24)(11,15)(13,23)(16,20), (2,17)(4,9)(6,19)(7,21)(11,15)(13,23), (2,7)(4,23)(6,15)(9,13)(11,19)(17,21), (1,5,3)(2,4,6)(7,13,19)(8,20,14)(9,15,21)(10,22,16)(11,17,23)(12,24,18), (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)(3,18)(4,13)(5,10)(6,11)(7,17)(8,22)(9,23)(12,16)(14,24)(15,19), (1,12)(2,17)(3,22)(4,9)(5,14)(6,19)(7,21)(8,18)(10,24)(11,15)(13,23)(16,20), (2,17)(4,9)(6,19)(7,21)(11,15)(13,23), (2,7)(4,23)(6,15)(9,13)(11,19)(17,21), (1,5,3)(2,4,6)(7,13,19)(8,20,14)(9,15,21)(10,22,16)(11,17,23)(12,24,18), (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),(3,18),(4,13),(5,10),(6,11),(7,17),(8,22),(9,23),(12,16),(14,24),(15,19)], [(1,12),(2,17),(3,22),(4,9),(5,14),(6,19),(7,21),(8,18),(10,24),(11,15),(13,23),(16,20)], [(2,17),(4,9),(6,19),(7,21),(11,15),(13,23)], [(2,7),(4,23),(6,15),(9,13),(11,19),(17,21)], [(1,5,3),(2,4,6),(7,13,19),(8,20,14),(9,15,21),(10,22,16),(11,17,23),(12,24,18)], [(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 13)(2 14)(3 8)(4 7)(5 19)(6 24)(9 11)(10 12)(15 17)(16 18)(20 22)(21 23)

(1 15)(2 18)(3 12)(4 9)(5 23)(6 20)(7 11)(8 10)(13 17)(14 16)(19 21)(22 24)

(2 18)(3 12)(5 23)(8 10)(14 16)(19 21)

(2 16)(3 10)(5 21)(8 12)(14 18)(19 23)

(1 6 4)(2 3 5)(7 15 22)(8 23 16)(9 17 24)(10 19 18)(11 13 20)(12 21 14)

(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)| (1,13)(2,14)(3,8)(4,7)(5,19)(6,24)(9,11)(10,12)(15,17)(16,18)(20,22)(21,23), (1,15)(2,18)(3,12)(4,9)(5,23)(6,20)(7,11)(8,10)(13,17)(14,16)(19,21)(22,24), (2,18)(3,12)(5,23)(8,10)(14,16)(19,21), (2,16)(3,10)(5,21)(8,12)(14,18)(19,23), (1,6,4)(2,3,5)(7,15,22)(8,23,16)(9,17,24)(10,19,18)(11,13,20)(12,21,14), (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( (1,13)(2,14)(3,8)(4,7)(5,19)(6,24)(9,11)(10,12)(15,17)(16,18)(20,22)(21,23), (1,15)(2,18)(3,12)(4,9)(5,23)(6,20)(7,11)(8,10)(13,17)(14,16)(19,21)(22,24), (2,18)(3,12)(5,23)(8,10)(14,16)(19,21), (2,16)(3,10)(5,21)(8,12)(14,18)(19,23), (1,6,4)(2,3,5)(7,15,22)(8,23,16)(9,17,24)(10,19,18)(11,13,20)(12,21,14), (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([[(1,13),(2,14),(3,8),(4,7),(5,19),(6,24),(9,11),(10,12),(15,17),(16,18),(20,22),(21,23)], [(1,15),(2,18),(3,12),(4,9),(5,23),(6,20),(7,11),(8,10),(13,17),(14,16),(19,21),(22,24)], [(2,18),(3,12),(5,23),(8,10),(14,16),(19,21)], [(2,16),(3,10),(5,21),(8,12),(14,18),(19,23)], [(1,6,4),(2,3,5),(7,15,22),(8,23,16),(9,17,24),(10,19,18),(11,13,20),(12,21,14)], [(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 A₄wrC₂ over Q

action	f(x)	Disc(f)
8T42	x⁸-2x⁷-12x⁶+16x⁵+52x⁴-26x³-80x²-24x-1	2¹⁸·5⁴·379²
12T126	x¹²-12x¹⁰+48x⁸-80x⁶+54x⁴-13x²+1	2¹²·5⁶·139⁴
12T128	x¹²-64x¹⁰-24x⁹+1450x⁸+1024x⁷-13572x⁶-13376x⁵+45009x⁴+58784x³-8716x²-28424x-6776	2⁶⁰·3⁴·7²·11²·103⁸·1481²·15451²·24113²
12T129	x¹²-62x¹⁰-48x⁹+1114x⁸+1984x⁷-5232x⁶-14872x⁵-4455x⁴+12320x³+8780x²+1380x-73	2²⁷·3²·103⁸·229²·3877²·4003²·2258791²

Matrix representation of A₄wrC₂ ►in GL₆(Z)

0	0	1	0	0	0
-1	-1	-1	0	0	0
1	0	0	0	0	0
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	-1	-1	-1

-1	-1	-1	0	0	0
0	0	1	0	0	0
0	1	0	0	0	0
0	0	0	-1	-1	-1
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	-1	-1	-1
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	-1	-1	-1
0	0	0	1	0	0

1	0	0	0	0	0
-1	-1	-1	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	-1	-1	-1

0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	-1	-1	-1
1	0	0	0	0	0
0	0	1	0	0	0
-1	-1	-1	0	0	0

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] >;

A₄wrC₂ 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 A₄wrC₂ in TeX
Character table of A₄wrC₂ in TeX

G = A4wrC2 order 288 = 25·32

Wreath product of A4 by C2

G = A₄wrC₂ order 288 = 2⁵·3²

Wreath product of A₄ by C₂