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## G = C3×C3⋊S4order 216 = 23·33

### Direct product of C3 and C3⋊S4

Aliases: C3×C3⋊S4, C628S3, C323S4, C3⋊(C3×S4), A4⋊(C3×S3), (C3×A4)⋊3S3, (C3×A4)⋊4C6, (C32×A4)⋊2C2, C22⋊(C3×C3⋊S3), (C2×C6)⋊2(C3×S3), (C2×C6)⋊1(C3⋊S3), SmallGroup(216,164)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C3×C3⋊S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C32×A4 — C3×C3⋊S4
 Lower central C3×A4 — C3×C3⋊S4
 Upper central C1 — C3

Generators and relations for C3×C3⋊S4
G = < a,b,c,d,e,f | a3=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 360 in 77 conjugacy classes, 18 normal (12 characteristic)
C1, C2 [×2], C3 [×2], C3 [×7], C4, C22, C22, S3 [×4], C6 [×4], D4, C32, C32 [×8], Dic3, C12, A4 [×3], A4 [×3], D6, C2×C6 [×2], C2×C6 [×2], C3×S3 [×4], C3⋊S3, C3×C6, C3⋊D4, C3×D4, S4 [×3], C33, C3×Dic3, C3×A4, C3×A4 [×3], C3×A4 [×4], S3×C6, C62, C3×C3⋊S3, C3×C3⋊D4, C3×S4 [×3], C3⋊S4, C32×A4, C3×C3⋊S4
Quotients: C1, C2, C3, S3 [×4], C6, C3×S3 [×4], C3⋊S3, S4, C3×C3⋊S3, C3×S4, C3⋊S4, C3×C3⋊S4

Character table of C3×C3⋊S4

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 4 6A 6B 6C 6D 6E 6F 6G 12A 12B size 1 3 18 1 1 2 2 2 8 8 8 8 8 8 8 8 8 18 3 3 6 6 6 18 18 18 18 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 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 -1 -1 -1 linear of order 2 ρ3 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ4 1 1 -1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ3 -1 ζ3 ζ32 ζ32 1 ζ3 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ5 1 1 -1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ32 -1 ζ32 ζ3 ζ3 1 ζ32 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ6 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 2 2 0 2 2 -1 -1 -1 -1 -1 -1 2 -1 2 -1 -1 2 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 2 -1 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ9 2 2 0 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 2 2 2 2 2 0 0 0 0 orthogonal lifted from S3 ρ10 2 2 0 2 2 -1 -1 -1 2 -1 2 -1 -1 -1 -1 2 -1 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ11 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 ζ65 -1+√-3 -1 -1 2 ζ6 -1-√-3 ζ6 ζ65 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×S3 ρ12 2 2 0 -1+√-3 -1-√-3 -1-√-3 -1+√-3 2 ζ65 ζ65 -1 -1 -1 ζ6 ζ6 ζ6 ζ65 0 -1+√-3 -1-√-3 -1-√-3 2 -1+√-3 0 0 0 0 complex lifted from C3×S3 ρ13 2 2 0 -1-√-3 -1+√-3 -1+√-3 -1-√-3 2 ζ6 ζ6 -1 -1 -1 ζ65 ζ65 ζ65 ζ6 0 -1-√-3 -1+√-3 -1+√-3 2 -1-√-3 0 0 0 0 complex lifted from C3×S3 ρ14 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 -1+√-3 ζ65 2 -1 -1 ζ6 ζ6 -1-√-3 ζ65 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×S3 ρ15 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 ζ6 ζ6 -1 2 -1 -1+√-3 ζ65 ζ65 -1-√-3 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×S3 ρ16 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 -1-√-3 ζ6 2 -1 -1 ζ65 ζ65 -1+√-3 ζ6 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×S3 ρ17 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 ζ6 -1-√-3 -1 -1 2 ζ65 -1+√-3 ζ65 ζ6 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×S3 ρ18 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 ζ65 ζ65 -1 2 -1 -1-√-3 ζ6 ζ6 -1+√-3 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×S3 ρ19 3 -1 -1 3 3 3 3 3 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 1 1 orthogonal lifted from S4 ρ20 3 -1 1 3 3 3 3 3 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 -1 -1 orthogonal lifted from S4 ρ21 3 -1 -1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 0 0 0 1 ζ65 ζ6 ζ6 -1 ζ65 ζ65 ζ6 ζ3 ζ32 complex lifted from C3×S4 ρ22 3 -1 1 -3+3√-3/2 -3-3√-3/2 -3-3√-3/2 -3+3√-3/2 3 0 0 0 0 0 0 0 0 0 -1 ζ65 ζ6 ζ6 -1 ζ65 ζ3 ζ32 ζ65 ζ6 complex lifted from C3×S4 ρ23 3 -1 1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 0 0 0 -1 ζ6 ζ65 ζ65 -1 ζ6 ζ32 ζ3 ζ6 ζ65 complex lifted from C3×S4 ρ24 3 -1 -1 -3-3√-3/2 -3+3√-3/2 -3+3√-3/2 -3-3√-3/2 3 0 0 0 0 0 0 0 0 0 1 ζ6 ζ65 ζ65 -1 ζ6 ζ6 ζ65 ζ32 ζ3 complex lifted from C3×S4 ρ25 6 -2 0 6 6 -3 -3 -3 0 0 0 0 0 0 0 0 0 0 -2 -2 1 1 1 0 0 0 0 orthogonal lifted from C3⋊S4 ρ26 6 -2 0 -3+3√-3 -3-3√-3 3+3√-3/2 3-3√-3/2 -3 0 0 0 0 0 0 0 0 0 0 1-√-3 1+√-3 ζ32 1 ζ3 0 0 0 0 complex faithful ρ27 6 -2 0 -3-3√-3 -3+3√-3 3-3√-3/2 3+3√-3/2 -3 0 0 0 0 0 0 0 0 0 0 1+√-3 1-√-3 ζ3 1 ζ32 0 0 0 0 complex faithful

Permutation representations of C3×C3⋊S4
On 24 points - transitive group 24T564
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 3 2)(4 5 6)(7 8 9)(10 12 11)(13 15 14)(16 17 18)(19 21 20)(22 23 24)
(1 14)(2 15)(3 13)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(10 20)(11 21)(12 19)
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(1 2 3)(4 16 22)(5 17 23)(6 18 24)(7 9 8)(10 14 19)(11 15 20)(12 13 21)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(24,564);

C3×C3⋊S4 is a maximal subgroup of   C3×S3×S4  C6210D6

Matrix representation of C3×C3⋊S4 in GL5(𝔽13)

 3 0 0 0 0 0 3 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 0 12 0 0 0 1 12 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 12 1 0 0 0 12 0 0 0 1 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 12 0 1 0 0 12 1 0
,
 12 1 0 0 0 12 0 0 0 0 0 0 1 0 12 0 0 0 0 12 0 0 0 1 12
,
 0 12 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(13))| [3,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[0,1,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[12,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,12,12],[0,12,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C3×C3⋊S4 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes S_4
% in TeX

G:=Group("C3xC3:S4");
// GroupNames label

G:=SmallGroup(216,164);
// by ID

G=gap.SmallGroup(216,164);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,218,867,3244,556,1949,989]);
// Polycyclic

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

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