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## G = C2×S3×A4order 144 = 24·32

### Direct product of C2, S3 and A4

Aliases: C2×S3×A4, C6⋊(C2×A4), C3⋊(C22×A4), (C22×C6)⋊C6, (C6×A4)⋊3C2, (S3×C23)⋊C3, (C22×S3)⋊C6, C222(S3×C6), C232(C3×S3), (C3×A4)⋊4C22, (C2×C6)⋊(C2×C6), SmallGroup(144,190)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×S3×A4
 Chief series C1 — C3 — C2×C6 — C3×A4 — S3×A4 — C2×S3×A4
 Lower central C2×C6 — C2×S3×A4
 Upper central C1 — C2

Generators and relations for C2×S3×A4
G = < a,b,c,d,e,f | a2=b3=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 336 in 82 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C22, C22 [×12], S3 [×2], S3 [×2], C6, C6 [×6], C23, C23 [×6], C32, A4, A4, D6, D6 [×9], C2×C6, C2×C6 [×3], C24, C3×S3 [×2], C3×C6, C2×A4, C2×A4 [×3], C22×S3 [×2], C22×S3 [×4], C22×C6, C3×A4, S3×C6, C22×A4, S3×C23, S3×A4 [×2], C6×A4, C2×S3×A4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], A4, D6, C2×C6, C3×S3, C2×A4 [×3], S3×C6, C22×A4, S3×A4, C2×S3×A4

Character table of C2×S3×A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K size 1 1 3 3 3 3 9 9 2 4 4 8 8 2 4 4 6 6 8 8 12 12 12 12 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 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 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 ρ5 1 -1 1 -1 1 -1 1 -1 1 ζ3 ζ32 ζ32 ζ3 -1 ζ6 ζ65 1 -1 ζ65 ζ6 ζ6 ζ65 ζ32 ζ3 linear of order 6 ρ6 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 1 1 -1 1 1 -1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ8 1 -1 -1 -1 1 1 -1 1 1 ζ3 ζ32 ζ32 ζ3 -1 ζ6 ζ65 1 -1 ζ65 ζ6 ζ32 ζ3 ζ6 ζ65 linear of order 6 ρ9 1 -1 -1 -1 1 1 -1 1 1 ζ32 ζ3 ζ3 ζ32 -1 ζ65 ζ6 1 -1 ζ6 ζ65 ζ3 ζ32 ζ65 ζ6 linear of order 6 ρ10 1 1 -1 1 1 -1 -1 -1 1 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ11 1 -1 1 -1 1 -1 1 -1 1 ζ32 ζ3 ζ3 ζ32 -1 ζ65 ζ6 1 -1 ζ6 ζ65 ζ65 ζ6 ζ3 ζ32 linear of order 6 ρ12 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ13 2 -2 0 -2 2 0 0 0 -1 2 2 -1 -1 1 -2 -2 -1 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ14 2 2 0 2 2 0 0 0 -1 2 2 -1 -1 -1 2 2 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ15 2 -2 0 -2 2 0 0 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 1 1-√-3 1+√-3 -1 1 ζ32 ζ3 0 0 0 0 complex lifted from S3×C6 ρ16 2 -2 0 -2 2 0 0 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 1 1+√-3 1-√-3 -1 1 ζ3 ζ32 0 0 0 0 complex lifted from S3×C6 ρ17 2 2 0 2 2 0 0 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 -1 -1-√-3 -1+√-3 -1 -1 ζ65 ζ6 0 0 0 0 complex lifted from C3×S3 ρ18 2 2 0 2 2 0 0 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 -1 -1+√-3 -1-√-3 -1 -1 ζ6 ζ65 0 0 0 0 complex lifted from C3×S3 ρ19 3 -3 3 1 -1 -3 -1 1 3 0 0 0 0 -3 0 0 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ20 3 3 -3 -1 -1 -3 1 1 3 0 0 0 0 3 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ21 3 3 3 -1 -1 3 -1 -1 3 0 0 0 0 3 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ22 3 -3 -3 1 -1 3 1 -1 3 0 0 0 0 -3 0 0 -1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ23 6 -6 0 2 -2 0 0 0 -3 0 0 0 0 3 0 0 1 -1 0 0 0 0 0 0 orthogonal faithful ρ24 6 6 0 -2 -2 0 0 0 -3 0 0 0 0 -3 0 0 1 1 0 0 0 0 0 0 orthogonal lifted from S3×A4

Permutation representations of C2×S3×A4
On 18 points - transitive group 18T60
Generators in S18
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 16)(14 18)(15 17)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)

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

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

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

G:=TransitiveGroup(18,60);

On 24 points - transitive group 24T250
Generators in S24
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 13)(2 15)(3 14)(4 16)(5 18)(6 17)(7 19)(8 21)(9 20)(10 22)(11 24)(12 23)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(4 7 10)(5 8 11)(6 9 12)(16 19 22)(17 20 23)(18 21 24)

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

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

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

G:=TransitiveGroup(24,250);

C2×S3×A4 is a maximal subgroup of   D6⋊S4  A4⋊D12
C2×S3×A4 is a maximal quotient of   SL2(𝔽3).11D6  Dic6.A4  D12.A4

Matrix representation of C2×S3×A4 in GL5(ℤ)

 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 0 -1 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 -1 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,-1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;

C2×S3×A4 in GAP, Magma, Sage, TeX

C_2\times S_3\times A_4
% in TeX

G:=Group("C2xS3xA4");
// GroupNames label

G:=SmallGroup(144,190);
// by ID

G=gap.SmallGroup(144,190);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-3,231,106,3461]);
// Polycyclic

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

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