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G = C6×S4order 144 = 24·32

Direct product of C6 and S4

direct product, non-abelian, soluble, monomial

Aliases: C6×S4, (C2×A4)⋊C6, A4⋊(C2×C6), (C2×C6)⋊2D6, C22⋊(S3×C6), C23⋊(C3×S3), (C6×A4)⋊1C2, (C22×C6)⋊1S3, (C3×A4)⋊2C22, SmallGroup(144,188)

Series: Derived Chief Lower central Upper central

C1C22A4 — C6×S4
C1C22A4C3×A4C3×S4 — C6×S4
A4 — C6×S4
C1C6

Generators and relations for C6×S4
 G = < a,b,c,d,e | a6=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Subgroups: 216 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×2], C22, C22 [×6], S3 [×2], C6, C6 [×6], C2×C4, D4 [×4], C23, C23, C32, C12 [×2], A4, A4, D6, C2×C6, C2×C6 [×6], C2×D4, C3×S3 [×2], C3×C6, C2×C12, C3×D4 [×4], S4 [×2], C2×A4, C2×A4, C22×C6, C22×C6, C3×A4, S3×C6, C6×D4, C2×S4, C3×S4 [×2], C6×A4, C6×S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C6×S4

Character table of C6×S4

 class 12A2B2C2D2E3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K6L6M12A12B12C12D
 size 113366118886611333366668886666
ρ1111111111111111111111111111111    trivial
ρ21111-1-111111-1-1111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ31-11-1-11111111-1-1-1-1-11111-1-1-1-1-11-11-1    linear of order 2
ρ41-11-11-111111-11-1-1-1-111-1-111-1-1-1-11-11    linear of order 2
ρ51111-1-1ζ3ζ32ζ321ζ3-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ6ζ65ζ65ζ6ζ321ζ3ζ6ζ65ζ65ζ6    linear of order 6
ρ61-11-11-1ζ3ζ32ζ321ζ3-11ζ6ζ65ζ65ζ6ζ32ζ3ζ6ζ65ζ3ζ32ζ6-1ζ65ζ6ζ3ζ65ζ32    linear of order 6
ρ71111-1-1ζ32ζ3ζ31ζ32-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ65ζ6ζ6ζ65ζ31ζ32ζ65ζ6ζ6ζ65    linear of order 6
ρ81-11-11-1ζ32ζ3ζ31ζ32-11ζ65ζ6ζ6ζ65ζ3ζ32ζ65ζ6ζ32ζ3ζ65-1ζ6ζ65ζ32ζ6ζ3    linear of order 6
ρ91-11-1-11ζ32ζ3ζ31ζ321-1ζ65ζ6ζ6ζ65ζ3ζ32ζ3ζ32ζ6ζ65ζ65-1ζ6ζ3ζ6ζ32ζ65    linear of order 6
ρ101-11-1-11ζ3ζ32ζ321ζ31-1ζ6ζ65ζ65ζ6ζ32ζ3ζ32ζ3ζ65ζ6ζ6-1ζ65ζ32ζ65ζ3ζ6    linear of order 6
ρ11111111ζ3ζ32ζ321ζ311ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ321ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ12111111ζ32ζ3ζ31ζ3211ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ31ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ132-22-20022-1-1-100-2-2-2-22200001110000    orthogonal lifted from D6
ρ1422220022-1-1-1002222220000-1-1-10000    orthogonal lifted from S3
ρ15222200-1--3-1+-3ζ65-1ζ600-1+-3-1--3-1--3-1+-3-1+-3-1--30000ζ65-1ζ60000    complex lifted from C3×S3
ρ16222200-1+-3-1--3ζ6-1ζ6500-1--3-1+-3-1+-3-1--3-1--3-1+-30000ζ6-1ζ650000    complex lifted from C3×S3
ρ172-22-200-1--3-1+-3ζ65-1ζ6001--31+-31+-31--3-1+-3-1--30000ζ31ζ320000    complex lifted from S3×C6
ρ182-22-200-1+-3-1--3ζ6-1ζ65001+-31--31--31+-3-1--3-1+-30000ζ321ζ30000    complex lifted from S3×C6
ρ1933-1-11133000-1-133-1-1-1-11111000-1-1-1-1    orthogonal lifted from S4
ρ2033-1-1-1-1330001133-1-1-1-1-1-1-1-10001111    orthogonal lifted from S4
ρ213-3-11-1133000-11-3-311-1-111-1-1000-11-11    orthogonal lifted from C2×S4
ρ223-3-111-1330001-1-3-311-1-1-1-1110001-11-1    orthogonal lifted from C2×S4
ρ233-3-111-1-3-3-3/2-3+3-3/20001-13-3-3/23+3-3/2ζ32ζ3ζ65ζ6ζ65ζ6ζ32ζ3000ζ3ζ6ζ32ζ65    complex faithful
ρ243-3-111-1-3+3-3/2-3-3-3/20001-13+3-3/23-3-3/2ζ3ζ32ζ6ζ65ζ6ζ65ζ3ζ32000ζ32ζ65ζ3ζ6    complex faithful
ρ2533-1-111-3-3-3/2-3+3-3/2000-1-1-3+3-3/2-3-3-3/2ζ6ζ65ζ65ζ6ζ3ζ32ζ32ζ3000ζ65ζ6ζ6ζ65    complex lifted from C3×S4
ρ2633-1-111-3+3-3/2-3-3-3/2000-1-1-3-3-3/2-3+3-3/2ζ65ζ6ζ6ζ65ζ32ζ3ζ3ζ32000ζ6ζ65ζ65ζ6    complex lifted from C3×S4
ρ273-3-11-11-3-3-3/2-3+3-3/2000-113-3-3/23+3-3/2ζ32ζ3ζ65ζ6ζ3ζ32ζ6ζ65000ζ65ζ32ζ6ζ3    complex faithful
ρ2833-1-1-1-1-3+3-3/2-3-3-3/200011-3-3-3/2-3+3-3/2ζ65ζ6ζ6ζ65ζ6ζ65ζ65ζ6000ζ32ζ3ζ3ζ32    complex lifted from C3×S4
ρ293-3-11-11-3+3-3/2-3-3-3/2000-113+3-3/23-3-3/2ζ3ζ32ζ6ζ65ζ32ζ3ζ65ζ6000ζ6ζ3ζ65ζ32    complex faithful
ρ3033-1-1-1-1-3-3-3/2-3+3-3/200011-3+3-3/2-3-3-3/2ζ6ζ65ζ65ζ6ζ65ζ6ζ6ζ65000ζ3ζ32ζ32ζ3    complex lifted from C3×S4

Permutation representations of C6×S4
On 18 points - transitive group 18T61
Generators in S18
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 16 9)(2 17 10)(3 18 11)(4 13 12)(5 14 7)(6 15 8)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)

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

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

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

G:=TransitiveGroup(18,61);

On 24 points - transitive group 24T254
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 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(7 19 17)(8 20 18)(9 21 13)(10 22 14)(11 23 15)(12 24 16)
(7 19)(8 20)(9 21)(10 22)(11 23)(12 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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16), (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (7,19,17)(8,20,18)(9,21,13)(10,22,14)(11,23,15)(12,24,16), (7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;

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

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

G:=TransitiveGroup(24,254);

C6×S4 is a maximal subgroup of   Dic3⋊S4  D6⋊S4

Matrix representation of C6×S4 in GL3(𝔽7) generated by

300
030
003
,
032
503
006
,
254
413
353
,
254
034
042
,
254
325
624
G:=sub<GL(3,GF(7))| [3,0,0,0,3,0,0,0,3],[0,5,0,3,0,0,2,3,6],[2,4,3,5,1,5,4,3,3],[2,0,0,5,3,4,4,4,2],[2,3,6,5,2,2,4,5,4] >;

C6×S4 in GAP, Magma, Sage, TeX

C_6\times S_4
% in TeX

G:=Group("C6xS4");
// GroupNames label

G:=SmallGroup(144,188);
// by ID

G=gap.SmallGroup(144,188);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,579,2164,202,1301,347]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Character table of C6×S4 in TeX

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