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## G = C3×C3.S4order 216 = 23·33

### Direct product of C3 and C3.S4

Aliases: C3×C3.S4, C32.2S4, C62.6S3, (C2×C6)⋊1D9, C22⋊(C3×D9), C3.A44C6, C3.2(C3×S4), (C3×C3.A4)⋊2C2, (C2×C6).3(C3×S3), SmallGroup(216,91)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C3×C3.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C3×C3.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=f2=1, e3=b, 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=b-1e2 >

Character table of C3×C3.S4

 class 1 2A 2B 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G 9A 9B 9C 9D 9E 9F 9G 9H 9I 12A 12B size 1 3 18 1 1 2 2 2 18 3 3 6 6 6 18 18 8 8 8 8 8 8 8 8 8 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 ζ32 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ32 linear of order 3 ρ5 1 1 -1 ζ3 ζ32 ζ32 ζ3 1 -1 ζ32 ζ3 ζ3 ζ32 1 ζ6 ζ65 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ65 ζ6 linear of order 6 ρ6 1 1 -1 ζ32 ζ3 ζ3 ζ32 1 -1 ζ3 ζ32 ζ32 ζ3 1 ζ65 ζ6 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ6 ζ65 linear of order 6 ρ7 2 2 0 2 2 2 2 2 0 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 2 -1 -1 -1 0 2 2 -1 -1 -1 0 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 0 0 orthogonal lifted from D9 ρ9 2 2 0 2 2 -1 -1 -1 0 2 2 -1 -1 -1 0 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 0 0 orthogonal lifted from D9 ρ10 2 2 0 2 2 -1 -1 -1 0 2 2 -1 -1 -1 0 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 0 0 orthogonal lifted from D9 ρ11 2 2 0 -1-√-3 -1+√-3 -1+√-3 -1-√-3 2 0 -1+√-3 -1-√-3 -1-√-3 -1+√-3 2 0 0 ζ6 ζ6 ζ65 ζ65 ζ65 -1 -1 -1 ζ6 0 0 complex lifted from C3×S3 ρ12 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 0 ζ92+ζ9 ζ98+ζ94 ζ94+ζ92 ζ95+ζ9 ζ98+ζ97 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ95 0 0 complex lifted from C3×D9 ρ13 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 0 ζ95+ζ9 ζ94+ζ92 ζ92+ζ9 ζ97+ζ95 ζ98+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ97 0 0 complex lifted from C3×D9 ρ14 2 2 0 -1+√-3 -1-√-3 -1-√-3 -1+√-3 2 0 -1-√-3 -1+√-3 -1+√-3 -1-√-3 2 0 0 ζ65 ζ65 ζ6 ζ6 ζ6 -1 -1 -1 ζ65 0 0 complex lifted from C3×S3 ρ15 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 0 ζ97+ζ95 ζ92+ζ9 ζ95+ζ9 ζ98+ζ97 ζ94+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ94 0 0 complex lifted from C3×D9 ρ16 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 0 ζ98+ζ97 ζ95+ζ9 ζ97+ζ95 ζ98+ζ94 ζ92+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ94+ζ92 0 0 complex lifted from C3×D9 ρ17 2 2 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 0 ζ94+ζ92 ζ98+ζ97 ζ98+ζ94 ζ92+ζ9 ζ97+ζ95 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ9 0 0 complex lifted from C3×D9 ρ18 2 2 0 -1-√-3 -1+√-3 ζ65 ζ6 -1 0 -1+√-3 -1-√-3 ζ6 ζ65 -1 0 0 ζ98+ζ94 ζ97+ζ95 ζ98+ζ97 ζ94+ζ92 ζ95+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ92+ζ9 0 0 complex lifted from C3×D9 ρ19 3 -1 1 3 3 3 3 3 -1 -1 -1 -1 -1 -1 1 1 0 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from S4 ρ20 3 -1 -1 3 3 3 3 3 1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 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 1 ζ65 ζ6 ζ6 ζ65 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 ζ32 ζ3 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 1 ζ6 ζ65 ζ65 ζ6 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 ζ3 ζ32 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 -1 ζ6 ζ65 ζ65 ζ6 -1 ζ32 ζ3 0 0 0 0 0 0 0 0 0 ζ65 ζ6 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 -1 ζ65 ζ6 ζ6 ζ65 -1 ζ3 ζ32 0 0 0 0 0 0 0 0 0 ζ6 ζ65 complex lifted from C3×S4 ρ25 6 -2 0 6 6 -3 -3 -3 0 -2 -2 1 1 1 0 0 0 0 0 0 0 0 0 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 1+√-3 1-√-3 ζ3 ζ32 1 0 0 0 0 0 0 0 0 0 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 1-√-3 1+√-3 ζ32 ζ3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C3×C3.S4
On 36 points
Generators in S36
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 34 31)(29 35 32)(30 36 33)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 27)(3 20)(4 21)(6 23)(7 24)(9 26)(11 30)(12 31)(14 33)(15 34)(17 36)(18 28)
(1 27)(2 19)(4 21)(5 22)(7 24)(8 25)(10 29)(12 31)(13 32)(15 34)(16 35)(18 28)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36)
(1 29)(2 28)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

G:=sub<Sym(36)| (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,27)(3,20)(4,21)(6,23)(7,24)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(18,28), (1,27)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(12,31)(13,32)(15,34)(16,35)(18,28), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36), (1,29)(2,28)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;

G:=Group( (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15)(19,22,25)(20,23,26)(21,24,27)(28,34,31)(29,35,32)(30,36,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,27)(3,20)(4,21)(6,23)(7,24)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(18,28), (1,27)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(12,31)(13,32)(15,34)(16,35)(18,28), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36), (1,29)(2,28)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );

G=PermutationGroup([[(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15),(19,22,25),(20,23,26),(21,24,27),(28,34,31),(29,35,32),(30,36,33)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(1,27),(3,20),(4,21),(6,23),(7,24),(9,26),(11,30),(12,31),(14,33),(15,34),(17,36),(18,28)], [(1,27),(2,19),(4,21),(5,22),(7,24),(8,25),(10,29),(12,31),(13,32),(15,34),(16,35),(18,28)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36)], [(1,29),(2,28),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])

Matrix representation of C3×C3.S4 in GL7(𝔽37)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 26 0 0 0 0 0 0 10 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36
,
 16 0 0 0 0 0 0 1 7 0 0 0 0 0 0 0 2 12 0 0 0 0 0 21 34 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 1 28 0 0 0 0 0 0 36 0 0 0 0 0 0 0 35 25 0 0 0 0 0 28 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0

G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[26,10,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[16,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,2,21,0,0,0,0,0,12,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,35,28,0,0,0,0,0,25,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;

C3×C3.S4 in GAP, Magma, Sage, TeX

C_3\times C_3.S_4
% in TeX

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

G:=SmallGroup(216,91);
// by ID

G=gap.SmallGroup(216,91);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,542,122,867,3244,556,1949,989]);
// Polycyclic

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

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