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G = C6xS4order 144 = 24·32

Direct product of C6 and S4

direct product, non-abelian, soluble, monomial

Aliases: C6xS4, (C2xA4):C6, A4:(C2xC6), (C2xC6):2D6, C22:(S3xC6), C23:(C3xS3), (C6xA4):1C2, (C22xC6):1S3, (C3xA4):2C22, SmallGroup(144,188)

Series: Derived Chief Lower central Upper central

C1C22A4 — C6xS4
C1C22A4C3xA4C3xS4 — C6xS4
A4 — C6xS4
C1C6

Generators and relations for C6xS4
 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, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, C12, A4, A4, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3xC6, C2xC12, C3xD4, S4, C2xA4, C2xA4, C22xC6, C22xC6, C3xA4, S3xC6, C6xD4, C2xS4, C3xS4, C6xA4, C6xS4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2xC6, C3xS3, S4, S3xC6, C2xS4, C3xS4, C6xS4

Character table of C6xS4

 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 C3xS3
ρ16222200-1+-3-1--3ζ6-1ζ6500-1--3-1+-3-1+-3-1--3-1--3-1+-30000ζ6-1ζ650000    complex lifted from C3xS3
ρ172-22-200-1--3-1+-3ζ65-1ζ6001--31+-31+-31--3-1+-3-1--30000ζ31ζ320000    complex lifted from S3xC6
ρ182-22-200-1+-3-1--3ζ6-1ζ65001+-31--31--31+-3-1--3-1+-30000ζ321ζ30000    complex lifted from S3xC6
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 C2xS4
ρ223-3-111-1330001-1-3-311-1-1-1-1110001-11-1    orthogonal lifted from C2xS4
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 C3xS4
ρ2633-1-111-3+3-3/2-3-3-3/2000-1-1-3-3-3/2-3+3-3/2ζ65ζ6ζ6ζ65ζ32ζ3ζ3ζ32000ζ6ζ65ζ65ζ6    complex lifted from C3xS4
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 C3xS4
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 C3xS4

Permutation representations of C6xS4
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 15 9)(2 16 10)(3 17 11)(4 18 12)(5 13 7)(6 14 8)
(1 4)(2 5)(3 6)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)

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

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

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

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

G:=TransitiveGroup(24,254);

C6xS4 is a maximal subgroup of   Dic3:S4  D6:S4

Matrix representation of C6xS4 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] >;

C6xS4 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 C6xS4 in TeX

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