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## G = C6×A4order 72 = 23·32

### Direct product of C6 and A4

Aliases: C6×A4, C23⋊C32, (C2×C6)⋊2C6, C22⋊(C3×C6), (C22×C6)⋊C3, SmallGroup(72,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×A4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4
 Lower central C22 — C6×A4
 Upper central C1 — C6

Generators and relations for C6×A4
G = < a,b,c,d | a6=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C6×A4

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L size 1 1 3 3 1 1 4 4 4 4 4 4 1 1 3 3 3 3 4 4 4 4 4 4 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 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ4 1 -1 -1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 -1 -1 -1 1 1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 linear of order 6 ρ5 1 -1 -1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ65 ζ6 ζ65 ζ32 ζ3 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ7 1 1 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 linear of order 3 ρ8 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ9 1 -1 -1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ65 ζ6 ζ65 ζ32 ζ3 ζ6 ζ6 -1 ζ65 ζ65 ζ6 -1 linear of order 6 ρ10 1 -1 -1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 -1 -1 -1 1 1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 linear of order 6 ρ11 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ12 1 -1 -1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ6 ζ65 ζ6 ζ3 ζ32 ζ65 ζ65 -1 ζ6 ζ6 ζ65 -1 linear of order 6 ρ13 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 linear of order 3 ρ14 1 -1 -1 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ32 ζ3 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ6 linear of order 6 ρ15 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 linear of order 3 ρ16 1 -1 -1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ6 ζ65 ζ6 ζ3 ζ32 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 linear of order 6 ρ17 1 1 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 linear of order 3 ρ18 1 -1 -1 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ3 ζ32 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ65 linear of order 6 ρ19 3 -3 1 -1 3 3 0 0 0 0 0 0 -3 -3 1 -1 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 3 3 -1 -1 3 3 0 0 0 0 0 0 3 3 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ21 3 -3 1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3-3√-3/2 3+3√-3/2 ζ3 ζ6 ζ65 ζ32 0 0 0 0 0 0 complex faithful ρ22 3 -3 1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3+3√-3/2 3-3√-3/2 ζ32 ζ65 ζ6 ζ3 0 0 0 0 0 0 complex faithful ρ23 3 3 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 ζ6 ζ65 ζ6 ζ65 0 0 0 0 0 0 complex lifted from C3×A4 ρ24 3 3 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 ζ65 ζ6 ζ65 ζ6 0 0 0 0 0 0 complex lifted from C3×A4

Permutation representations of C6×A4
On 18 points - transitive group 18T25
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 15 11)(2 16 12)(3 17 7)(4 18 8)(5 13 9)(6 14 10)

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

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

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

G:=TransitiveGroup(18,25);

On 24 points - transitive group 24T71
Generators in S24
(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 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
(1 5 3)(2 6 4)(7 23 13)(8 24 14)(9 19 15)(10 20 16)(11 21 17)(12 22 18)

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

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

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

G:=TransitiveGroup(24,71);

C6×A4 is a maximal subgroup of   C6.7S4

Matrix representation of C6×A4 in GL3(𝔽7) generated by

 3 0 0 0 3 0 0 0 3
,
 6 0 0 4 0 1 4 1 0
,
 0 5 2 3 0 6 0 0 6
,
 2 0 4 0 0 5 0 2 5
G:=sub<GL(3,GF(7))| [3,0,0,0,3,0,0,0,3],[6,4,4,0,0,1,0,1,0],[0,3,0,5,0,0,2,6,6],[2,0,0,0,0,2,4,5,5] >;

C6×A4 in GAP, Magma, Sage, TeX

C_6\times A_4
% in TeX

G:=Group("C6xA4");
// GroupNames label

G:=SmallGroup(72,47);
// by ID

G=gap.SmallGroup(72,47);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,2,368,684]);
// Polycyclic

G:=Group<a,b,c,d|a^6=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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