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G = C6×A4order 72 = 23·32

Direct product of C6 and A4

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

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

Series: Derived Chief Lower central Upper central

C1C22 — C6×A4
C1C22C2×C6C3×A4 — C6×A4
C22 — C6×A4
C1C6

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 >

3C2
3C2
4C3
4C3
4C3
3C22
3C22
3C6
3C6
4C6
4C6
4C6
4C32
3C2×C6
3C2×C6
4C3×C6

Character table of C6×A4

 class 12A2B2C3A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H6I6J6K6L
 size 113311444444113333444444
ρ1111111111111111111111111    trivial
ρ21-1-1111111111-1-1-111-1-1-1-1-1-1-1    linear of order 2
ρ31111ζ3ζ321ζ3ζ321ζ3ζ32ζ3ζ32ζ3ζ32ζ3ζ321ζ3ζ321ζ3ζ32    linear of order 3
ρ41-1-1111ζ32ζ32ζ32ζ3ζ3ζ3-1-1-111-1ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ51-1-11ζ3ζ32ζ3ζ321ζ321ζ3ζ65ζ6ζ65ζ32ζ3ζ6ζ65ζ6-1ζ6-1ζ65    linear of order 6
ρ61111ζ3ζ32ζ3ζ321ζ321ζ3ζ3ζ32ζ3ζ32ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ71111ζ3ζ32ζ321ζ3ζ3ζ321ζ3ζ32ζ3ζ32ζ3ζ32ζ321ζ3ζ3ζ321    linear of order 3
ρ81111ζ32ζ31ζ32ζ31ζ32ζ3ζ32ζ3ζ32ζ3ζ32ζ31ζ32ζ31ζ32ζ3    linear of order 3
ρ91-1-11ζ3ζ32ζ321ζ3ζ3ζ321ζ65ζ6ζ65ζ32ζ3ζ6ζ6-1ζ65ζ65ζ6-1    linear of order 6
ρ101-1-1111ζ3ζ3ζ3ζ32ζ32ζ32-1-1-111-1ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ111111ζ32ζ3ζ32ζ31ζ31ζ32ζ32ζ3ζ32ζ3ζ32ζ3ζ32ζ31ζ31ζ32    linear of order 3
ρ121-1-11ζ32ζ3ζ31ζ32ζ32ζ31ζ6ζ65ζ6ζ3ζ32ζ65ζ65-1ζ6ζ6ζ65-1    linear of order 6
ρ13111111ζ32ζ32ζ32ζ3ζ3ζ3111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ141-1-11ζ3ζ321ζ3ζ321ζ3ζ32ζ65ζ6ζ65ζ32ζ3ζ6-1ζ65ζ6-1ζ65ζ6    linear of order 6
ρ15111111ζ3ζ3ζ3ζ32ζ32ζ32111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ161-1-11ζ32ζ3ζ32ζ31ζ31ζ32ζ6ζ65ζ6ζ3ζ32ζ65ζ6ζ65-1ζ65-1ζ6    linear of order 6
ρ171111ζ32ζ3ζ31ζ32ζ32ζ31ζ32ζ3ζ32ζ3ζ32ζ3ζ31ζ32ζ32ζ31    linear of order 3
ρ181-1-11ζ32ζ31ζ32ζ31ζ32ζ3ζ6ζ65ζ6ζ3ζ32ζ65-1ζ6ζ65-1ζ6ζ65    linear of order 6
ρ193-31-133000000-3-31-1-11000000    orthogonal lifted from C2×A4
ρ2033-1-13300000033-1-1-1-1000000    orthogonal lifted from A4
ρ213-31-1-3+3-3/2-3-3-3/20000003-3-3/23+3-3/2ζ3ζ6ζ65ζ32000000    complex faithful
ρ223-31-1-3-3-3/2-3+3-3/20000003+3-3/23-3-3/2ζ32ζ65ζ6ζ3000000    complex faithful
ρ2333-1-1-3-3-3/2-3+3-3/2000000-3-3-3/2-3+3-3/2ζ6ζ65ζ6ζ65000000    complex lifted from C3×A4
ρ2433-1-1-3+3-3/2-3-3-3/2000000-3+3-3/2-3-3-3/2ζ65ζ6ζ65ζ6000000    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 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(1 5 3)(2 6 4)(7 15 19)(8 16 20)(9 17 21)(10 18 22)(11 13 23)(12 14 24)

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

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

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

G:=TransitiveGroup(24,71);

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

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

300
030
003
,
600
401
410
,
052
306
006
,
204
005
025
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

Export

Subgroup lattice of C6×A4 in TeX
Character table of C6×A4 in TeX

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