A4xC3^2:C4 - GroupNames

G = A₄×C₃²⋊C₄ order 432 = 2⁴·3³

Direct product of A₄ and C₃²⋊C₄

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

Aliases: A₄×C₃²⋊C₄, C₆²⋊C₁₂, C₃²⋊₂(C₄×A₄), (C₃²×A₄)⋊₂C₄, C₂²⋊(C₃×C₃²⋊C₄), (A₄×C₃⋊S₃).₂C₂, C₃⋊S₃.₂(C₂×A₄), (C₂²×C₃²⋊C₄)⋊C₃, (C₂²×C₃⋊S₃).₂C₆, SmallGroup(432,744)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₆² — A₄×C₃²⋊C₄

Chief series

C₁ — C₃² — C₆² — C₂²×C₃⋊S₃ — A₄×C₃⋊S₃ — A₄×C₃²⋊C₄

Lower central

C₆² — A₄×C₃²⋊C₄

Upper central

C₁

Generators and relations for A₄×C₃²⋊C₄
G = < a,b,c,d,e,f | a²=b²=c³=d³=e³=f⁴=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, fef^-1=de=ed, fdf^-1=d^-1e >

Subgroups: 652 in 70 conjugacy classes, 12 normal (all characteristic)
C₁, C₂, C₃, C₄, C₂², C₂², S₃, C₆, C₂×C₄, C₂³, C₃², C₃², C₁₂, A₄, A₄, D₆, C₂×C₆, C₂²×C₄, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₂×A₄, C₂²×S₃, C₃³, C₃²⋊C₄, C₃²⋊C₄, C₃×A₄, C₂×C₃⋊S₃, C₆², C₄×A₄, C₃×C₃⋊S₃, S₃×A₄, C₂×C₃²⋊C₄, C₂²×C₃⋊S₃, C₃×C₃²⋊C₄, C₃²×A₄, C₂²×C₃²⋊C₄, A₄×C₃⋊S₃, A₄×C₃²⋊C₄
Quotients: C₁, C₂, C₃, C₄, C₆, C₁₂, A₄, C₂×A₄, C₃²⋊C₄, C₄×A₄, C₃×C₃²⋊C₄, A₄×C₃²⋊C₄

Character table of A₄×C₃²⋊C₄

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	3G	3H	4A	4B	4C	4D	6A	6B	6C	6D	12A	12B	12C	12D
size	1	3	9	27	4	4	4	4	16	16	16	16	9	9	27	27	12	12	36	36	36	36	36	36
ρ₁	1	1	1	1	1	1	1	1	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	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	ζ₃²	1	ζ₃	1	ζ₃²	ζ₃	ζ₃	ζ₃²	-1	-1	-1	-1	1	1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	linear of order 6
ρ₄	1	1	1	1	ζ₃	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₃²	linear of order 3
ρ₅	1	1	1	1	ζ₃	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃²	ζ₃	-1	-1	-1	-1	1	1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	linear of order 6
ρ₆	1	1	1	1	ζ₃²	1	ζ₃	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₃	linear of order 3
ρ₇	1	1	-1	-1	1	1	1	1	1	1	1	1	-i	i	i	-i	1	1	-1	-1	-i	-i	i	i	linear of order 4
ρ₈	1	1	-1	-1	1	1	1	1	1	1	1	1	i	-i	-i	i	1	1	-1	-1	i	i	-i	-i	linear of order 4
ρ₉	1	1	-1	-1	ζ₃²	1	ζ₃	1	ζ₃²	ζ₃	ζ₃	ζ₃²	i	-i	-i	i	1	1	ζ₆⁵	ζ₆	ζ₄ζ₃²	ζ₄ζ₃	ζ₄³ζ₃²	ζ₄³ζ₃	linear of order 12
ρ₁₀	1	1	-1	-1	ζ₃	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃²	ζ₃	i	-i	-i	i	1	1	ζ₆	ζ₆⁵	ζ₄ζ₃	ζ₄ζ₃²	ζ₄³ζ₃	ζ₄³ζ₃²	linear of order 12
ρ₁₁	1	1	-1	-1	ζ₃²	1	ζ₃	1	ζ₃²	ζ₃	ζ₃	ζ₃²	-i	i	i	-i	1	1	ζ₆⁵	ζ₆	ζ₄³ζ₃²	ζ₄³ζ₃	ζ₄ζ₃²	ζ₄ζ₃	linear of order 12
ρ₁₂	1	1	-1	-1	ζ₃	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃²	ζ₃	-i	i	i	-i	1	1	ζ₆	ζ₆⁵	ζ₄³ζ₃	ζ₄³ζ₃²	ζ₄ζ₃	ζ₄ζ₃²	linear of order 12
ρ₁₃	3	-1	3	-1	0	3	0	3	0	0	0	0	3	3	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from A₄
ρ₁₄	3	-1	3	-1	0	3	0	3	0	0	0	0	-3	-3	1	1	-1	-1	0	0	0	0	0	0	orthogonal lifted from C₂×A₄
ρ₁₅	3	-1	-3	1	0	3	0	3	0	0	0	0	3i	-3i	i	-i	-1	-1	0	0	0	0	0	0	complex lifted from C₄×A₄
ρ₁₆	3	-1	-3	1	0	3	0	3	0	0	0	0	-3i	3i	-i	i	-1	-1	0	0	0	0	0	0	complex lifted from C₄×A₄
ρ₁₇	4	4	0	0	4	1	4	-2	-2	1	-2	1	0	0	0	0	1	-2	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₁₈	4	4	0	0	4	-2	4	1	1	-2	1	-2	0	0	0	0	-2	1	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₄
ρ₁₉	4	4	0	0	-2-2√-3	-2	-2+2√-3	1	ζ₃²	1-√-3	ζ₃	1+√-3	0	0	0	0	-2	1	0	0	0	0	0	0	complex lifted from C₃×C₃²⋊C₄
ρ₂₀	4	4	0	0	-2+2√-3	1	-2-2√-3	-2	1-√-3	ζ₃²	1+√-3	ζ₃	0	0	0	0	1	-2	0	0	0	0	0	0	complex lifted from C₃×C₃²⋊C₄
ρ₂₁	4	4	0	0	-2-2√-3	1	-2+2√-3	-2	1+√-3	ζ₃	1-√-3	ζ₃²	0	0	0	0	1	-2	0	0	0	0	0	0	complex lifted from C₃×C₃²⋊C₄
ρ₂₂	4	4	0	0	-2+2√-3	-2	-2-2√-3	1	ζ₃	1+√-3	ζ₃²	1-√-3	0	0	0	0	-2	1	0	0	0	0	0	0	complex lifted from C₃×C₃²⋊C₄
ρ₂₃	12	-4	0	0	0	-6	0	3	0	0	0	0	0	0	0	0	2	-1	0	0	0	0	0	0	orthogonal faithful
ρ₂₄	12	-4	0	0	0	3	0	-6	0	0	0	0	0	0	0	0	-1	2	0	0	0	0	0	0	orthogonal faithful

Permutation representations of A₄×C₃²⋊C₄
►On 24 points - transitive group 24T1337

Generators in S₂₄

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

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

(1 6 7)(2 5 8)(9 24 18)(10 21 19)(11 22 20)(12 23 17)

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

(2 11 9)(3 13 15)(5 22 24)(8 20 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)| (1,4)(2,3)(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,7)(2,8)(3,5)(4,6)(9,18)(10,19)(11,20)(12,17)(13,22)(14,23)(15,24)(16,21), (1,6,7)(2,5,8)(9,24,18)(10,21,19)(11,22,20)(12,23,17), (1,10,12)(2,9,11)(3,15,13)(4,16,14)(5,24,22)(6,21,23)(7,19,17)(8,18,20), (2,11,9)(3,13,15)(5,22,24)(8,20,18), (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( (1,4)(2,3)(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22), (1,7)(2,8)(3,5)(4,6)(9,18)(10,19)(11,20)(12,17)(13,22)(14,23)(15,24)(16,21), (1,6,7)(2,5,8)(9,24,18)(10,21,19)(11,22,20)(12,23,17), (1,10,12)(2,9,11)(3,15,13)(4,16,14)(5,24,22)(6,21,23)(7,19,17)(8,18,20), (2,11,9)(3,13,15)(5,22,24)(8,20,18), (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([[(1,4),(2,3),(5,8),(6,7),(9,15),(10,16),(11,13),(12,14),(17,23),(18,24),(19,21),(20,22)], [(1,7),(2,8),(3,5),(4,6),(9,18),(10,19),(11,20),(12,17),(13,22),(14,23),(15,24),(16,21)], [(1,6,7),(2,5,8),(9,24,18),(10,21,19),(11,22,20),(12,23,17)], [(1,10,12),(2,9,11),(3,15,13),(4,16,14),(5,24,22),(6,21,23),(7,19,17),(8,18,20)], [(2,11,9),(3,13,15),(5,22,24),(8,20,18)], [(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(24,1337);

Matrix representation of A₄×C₃²⋊C₄ ►in GL₇(𝔽₁₃)

12	0	0	0	0	0	0
12	0	1	0	0	0	0
12	1	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

0	12	1	0	0	0	0
0	12	0	0	0	0	0
1	12	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

0	0	1	0	0	0	0
1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	12	12	0	0
0	0	0	1	0	0	0
0	0	0	0	0	12	12
0	0	0	0	0	1	0

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	0	1
0	0	0	0	0	12	12

5	0	0	0	0	0	0
0	5	0	0	0	0	0
0	0	5	0	0	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1
0	0	0	1	0	0	0
0	0	0	12	12	0	0

G:=sub<GL(7,GF(13))| [12,12,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,1,12],[5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,0,1,0,0] >;

A₄×C₃²⋊C₄ in GAP, Magma, Sage, TeX

A_4\times C_3^2\rtimes C_4

% in TeX

G:=Group("A4xC3^2:C4");

// GroupNames label

G:=SmallGroup(432,744);

// by ID

G=gap.SmallGroup(432,744);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,2,-3,3,42,514,221,10085,691,9414,2372]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^3=f^4=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*e*f^-1=d*e=e*d,f*d*f^-1=d^-1*e>;

// generators/relations

Export

Character table of A₄×C₃²⋊C₄ in TeX

G = A4×C32⋊C4 order 432 = 24·33

Direct product of A4 and C32⋊C4

G = A₄×C₃²⋊C₄ order 432 = 2⁴·3³

Direct product of A₄ and C₃²⋊C₄