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G = C3×A4order 36 = 22·32

Direct product of C3 and A4

Aliases: C3×A4, C22⋊C32, (C2×C6)⋊C3, SmallGroup(36,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4
 Chief series C1 — C22 — A4 — C3×A4
 Lower central C22 — C3×A4
 Upper central C1 — C3

Generators and relations for C3×A4
G = < a,b,c,d | a3=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C3×A4

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

Permutation representations of C3×A4
On 12 points - transitive group 12T20
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 2 3)(4 11 7)(5 12 8)(6 10 9)

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

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

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

G:=TransitiveGroup(12,20);

On 18 points - transitive group 18T8
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)
(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)
(1 16 10)(2 17 11)(3 18 12)(4 13 7)(5 14 8)(6 15 9)

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

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

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

G:=TransitiveGroup(18,8);

C3×A4 is a maximal subgroup of   C3⋊S4  C9⋊A4  C32⋊A4
C3×A4 is a maximal quotient of   C9⋊A4  C32.A4  C32⋊A4

Polynomial with Galois group C3×A4 over ℚ
actionf(x)Disc(f)
12T20x12-3x11-30x10+73x9+303x8-603x7-1171x6+2034x5+1254x4-2511x3+420x2+228x+8224·316·76·1636·61257412

Matrix representation of C3×A4 in GL3(𝔽7) generated by

 4 0 0 0 4 0 0 0 4
,
 0 6 1 0 6 0 1 6 0
,
 6 0 0 6 0 1 6 1 0
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(7))| [4,0,0,0,4,0,0,0,4],[0,0,1,6,6,6,1,0,0],[6,6,6,0,0,1,0,1,0],[0,0,1,1,0,0,0,1,0] >;

C3×A4 in GAP, Magma, Sage, TeX

C_3\times A_4
% in TeX

G:=Group("C3xA4");
// GroupNames label

G:=SmallGroup(36,11);
// by ID

G=gap.SmallGroup(36,11);
# by ID

G:=PCGroup([4,-3,-3,-2,2,218,435]);
// Polycyclic

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

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