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## G = A4×C32⋊C4order 432 = 24·33

### Direct product of A4 and C32⋊C4

Aliases: A4×C32⋊C4, C62⋊C12, C322(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

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