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## G = A42order 144 = 24·32

### Direct product of A4 and A4

Aliases: A42, PΩ+4(𝔽3), C24⋊C32, C22⋊A4⋊C3, (C22×A4)⋊C3, C221(C3×A4), SmallGroup(144,184)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — A42
 Chief series C1 — C22 — C24 — C22×A4 — A42
 Lower central C24 — A42
 Upper central C1

Generators and relations for A42
G = < a,b,c,d,e,f | a2=b2=c3=d2=e2=f3=1, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

3C2
3C2
9C2
4C3
4C3
16C3
16C3
3C22
3C22
9C22
9C22
9C22
12C6
12C6
16C32
3C23
3C23
9C23
4A4
4A4
4A4
4A4
12A4
12A4

Character table of A42

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D size 1 3 3 9 4 4 4 4 16 16 16 16 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ3 1 1 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 linear of order 3 ρ4 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 1 1 1 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 1 linear of order 3 ρ7 1 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 linear of order 3 ρ8 1 1 1 1 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ9 1 1 1 1 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 1 linear of order 3 ρ10 3 3 -1 -1 3 0 0 3 0 0 0 0 0 -1 -1 0 orthogonal lifted from A4 ρ11 3 -1 3 -1 0 3 3 0 0 0 0 0 -1 0 0 -1 orthogonal lifted from A4 ρ12 3 -1 3 -1 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 ζ65 0 0 ζ6 complex lifted from C3×A4 ρ13 3 3 -1 -1 -3-3√-3/2 0 0 -3+3√-3/2 0 0 0 0 0 ζ65 ζ6 0 complex lifted from C3×A4 ρ14 3 3 -1 -1 -3+3√-3/2 0 0 -3-3√-3/2 0 0 0 0 0 ζ6 ζ65 0 complex lifted from C3×A4 ρ15 3 -1 3 -1 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 ζ6 0 0 ζ65 complex lifted from C3×A4 ρ16 9 -3 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of A42
On 12 points - transitive group 12T85
Generators in S12
```(1 4)(2 12)(3 7)(5 9)(6 10)(8 11)
(1 8)(2 5)(3 10)(4 11)(6 7)(9 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 11)(2 12)(3 10)(4 8)(5 9)(6 7)
(1 2 3)(4 12 7)(5 10 8)(6 11 9)```

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

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

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

`G:=TransitiveGroup(12,85);`

On 16 points - transitive group 16T414
Generators in S16
```(1 15)(2 7)(3 13)(4 8)(5 6)(9 10)(11 12)(14 16)
(1 16)(2 5)(3 11)(4 9)(6 7)(8 10)(12 13)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 2)(3 4)(5 16)(6 14)(7 15)(8 13)(9 11)(10 12)
(1 4)(2 3)(5 11)(6 12)(7 13)(8 15)(9 16)(10 14)
(2 4 3)(5 9 11)(6 10 12)(7 8 13)```

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

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

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

`G:=TransitiveGroup(16,414);`

On 18 points - transitive group 18T62
Generators in S18
```(1 17)(2 18)(4 7)(6 9)(10 15)(11 13)
(2 18)(3 16)(4 7)(5 8)(11 13)(12 14)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(4 7)(5 8)(6 9)(10 15)(11 13)(12 14)
(1 17)(2 18)(3 16)(4 7)(5 8)(6 9)
(1 10 9)(2 11 7)(3 12 8)(4 18 13)(5 16 14)(6 17 15)```

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

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

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

`G:=TransitiveGroup(18,62);`

On 24 points - transitive group 24T212
Generators in S24
```(1 23)(2 24)(4 8)(5 9)(10 15)(12 14)(16 19)(17 20)
(2 24)(3 22)(5 9)(6 7)(10 15)(11 13)(17 20)(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)
(1 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(4 16 14)(5 17 15)(6 18 13)(7 21 11)(8 19 12)(9 20 10)```

`G:=sub<Sym(24)| (1,23)(2,24)(4,8)(5,9)(10,15)(12,14)(16,19)(17,20), (2,24)(3,22)(5,9)(6,7)(10,15)(11,13)(17,20)(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), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (4,16,14)(5,17,15)(6,18,13)(7,21,11)(8,19,12)(9,20,10)>;`

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

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

`G:=TransitiveGroup(24,212);`

A42 is a maximal subgroup of   A4≀C2  PSO4+ (𝔽3)
A42 is a maximal quotient of   Ω4+ (𝔽3)  C3.A42  C24⋊He3  C24⋊3- 1+2  C2423- 1+2

Polynomial with Galois group A42 over ℚ
actionf(x)Disc(f)
12T85x12-3x11-3x10+15x9-15x8-33x7+29x6+15x5-30x4-128x3-30x2+198x+48244·322·78·3792

Matrix representation of A42 in GL6(𝔽7)

 0 6 1 0 0 0 0 6 0 0 0 0 1 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 6 0 1 0 0 0 6 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 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 6 6 6 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 6 6 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 2 0 0 0 0 0 0 2 0 0 0 2 0 0

`G:=sub<GL(6,GF(7))| [0,0,1,0,0,0,6,6,6,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,6,6,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,1,0,0,0,0,6,0,0,0,0,1,6,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,6,0,1,0,0,0,6,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,0,2,0] >;`

A42 in GAP, Magma, Sage, TeX

`A_4^2`
`% in TeX`

`G:=Group("A4^2");`
`// GroupNames label`

`G:=SmallGroup(144,184);`
`// by ID`

`G=gap.SmallGroup(144,184);`
`# by ID`

`G:=PCGroup([6,-3,-3,-2,2,-2,2,170,81,3244,1301]);`
`// Polycyclic`

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

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