A4^2 - GroupNames

G = A₄² order 144 = 2⁴·3²

Direct product of A₄ and A₄

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄², PΩ+₄(F₃), C₂⁴:C₃², C₂²:A₄:C₃, (C₂²xA₄):C₃, C₂²:₁(C₃xA₄), SmallGroup(144,184)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — A₄²

Chief series

C₁ — C₂² — C₂⁴ — C₂²xA₄ — 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 >

Subgroups: 216 in 41 conjugacy classes, 11 normal (3 characteristic)
Quotients: C₁, C₃, C₃², A₄, C₃xA₄, A₄²

C₁

₃C₂

₉C₂

₄C₃

₁₆C₃

C₂²

₃C₂²

₉C₂²

₁₂C₆

₁₆C₃²

₃C₂³

₉C₂³

A₄

₄A₄

₄C₂xC₆

₄A₄

₄C₂xC₆

₄A₄

₁₂A₄

C₂⁴

₃C₂xA₄

₄C₃xA₄

C₂²:A₄

C₂²xA₄

A₄²

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 C₃xA₄
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 C₃xA₄
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 C₃xA₄
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 C₃xA₄
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 7)(2 15)(3 13)(4 8)(5 6)(9 10)(11 12)(14 16)

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

(1 4 3)(5 9 11)(6 10 12)(7 8 13)

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

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

G=PermutationGroup([[(1,7),(2,15),(3,13),(4,8),(5,6),(9,10),(11,12),(14,16)], [(1,5),(2,16),(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)], [(1,3),(2,4),(5,11),(6,12),(7,13),(8,15),(9,16),(10,14)], [(1,4,3),(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 A₄wrC₂ PSO₄⁺ (F₃)
A₄² is a maximal quotient of Ω₄⁺ (F₃) C₃.A₄² C₂⁴:He₃ C₂⁴:3_-¹⁺² C₂⁴:₂3_-¹⁺²

Polynomial with Galois group A₄² over Q

action	f(x)	Disc(f)
12T85	x¹²-3x¹¹-3x¹⁰+15x⁹-15x⁸-33x⁷+29x⁶+15x⁵-30x⁴-128x³-30x²+198x+48	2⁴⁴·3²²·7⁸·379²

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

Export

Subgroup lattice of A₄² in TeX
Character table of A₄² in TeX

G = A42 order 144 = 24·32

Direct product of A4 and A4

G = A₄² order 144 = 2⁴·3²

Direct product of A₄ and A₄