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## G = S3×C3.S4order 432 = 24·33

### Direct product of S3 and C3.S4

Aliases: S3×C3.S4, C62.17D6, (C2×C6)⋊D18, (C3×S3).S4, C3.3(S3×S4), C3.A41D6, (C22×S3)⋊D9, C222(S3×D9), C32.3S4⋊C2, C32.2(C2×S4), (S3×C2×C6).S3, (C3×C3.S4)⋊C2, (S3×C3.A4)⋊C2, (C2×C6).3S32, C31(C2×C3.S4), (C3×C3.A4)⋊C22, SmallGroup(432,522)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C3.A4 — S3×C3.S4
 Chief series C1 — C22 — C2×C6 — C62 — C3×C3.A4 — S3×C3.A4 — S3×C3.S4
 Lower central C3×C3.A4 — S3×C3.S4
 Upper central C1

Generators and relations for S3×C3.S4
G = < a,b,c,d,e,f,g | a3=b2=c3=d2=e2=g2=1, f3=c, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=c-1f2 >

Subgroups: 1120 in 123 conjugacy classes, 19 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, S3, C6, C2×C4, D4, C23, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×S3, C22×C6, C3×C9, C3.A4, C3.A4, D18, C3×Dic3, C3⋊Dic3, S32, S3×C6, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, C3×D9, S3×C9, C9⋊S3, C3.S4, C3.S4, C2×C3.A4, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C327D4, C2×S32, S3×C2×C6, S3×D9, C3×C3.A4, C2×C3.S4, S3×C3⋊D4, C3×C3.S4, C32.3S4, S3×C3.A4, S3×C3.S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, S32, C2×S4, C3.S4, S3×D9, C2×C3.S4, S3×S4, S3×C3.S4

Character table of S3×C3.S4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 9A 9B 9C 9D 9E 9F 12 18A 18B 18C size 1 3 3 9 18 54 2 2 4 18 54 6 6 6 12 18 36 8 8 8 16 16 16 36 24 24 24 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 -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 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 -2 0 -1 2 -1 -2 0 0 -1 2 -1 0 1 2 2 2 -1 -1 -1 1 0 0 0 orthogonal lifted from D6 ρ6 2 2 0 0 2 0 -1 2 -1 2 0 0 -1 2 -1 0 -1 2 2 2 -1 -1 -1 -1 0 0 0 orthogonal lifted from S3 ρ7 2 2 2 2 0 0 2 2 2 0 0 2 2 2 2 2 0 -1 -1 -1 -1 -1 -1 0 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 -2 -2 0 0 2 2 2 0 0 -2 2 2 2 -2 0 -1 -1 -1 -1 -1 -1 0 1 1 1 orthogonal lifted from D6 ρ9 2 2 2 2 0 0 2 -1 -1 0 0 -1 2 -1 -1 -1 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ10 2 2 2 2 0 0 2 -1 -1 0 0 -1 2 -1 -1 -1 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 2 -2 -2 0 0 2 -1 -1 0 0 1 2 -1 -1 1 0 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 0 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ12 2 2 -2 -2 0 0 2 -1 -1 0 0 1 2 -1 -1 1 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 0 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ13 2 2 -2 -2 0 0 2 -1 -1 0 0 1 2 -1 -1 1 0 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 0 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ14 2 2 2 2 0 0 2 -1 -1 0 0 -1 2 -1 -1 -1 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 0 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ15 3 -1 3 -1 -1 -1 3 3 3 1 1 3 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 orthogonal lifted from S4 ρ16 3 -1 3 -1 1 1 3 3 3 -1 -1 3 -1 -1 -1 -1 1 0 0 0 0 0 0 -1 0 0 0 orthogonal lifted from S4 ρ17 3 -1 -3 1 -1 1 3 3 3 1 -1 -3 -1 -1 -1 1 -1 0 0 0 0 0 0 1 0 0 0 orthogonal lifted from C2×S4 ρ18 3 -1 -3 1 1 -1 3 3 3 -1 1 -3 -1 -1 -1 1 1 0 0 0 0 0 0 -1 0 0 0 orthogonal lifted from C2×S4 ρ19 4 4 0 0 0 0 -2 4 -2 0 0 0 -2 4 -2 0 0 -2 -2 -2 1 1 1 0 0 0 0 orthogonal lifted from S32 ρ20 4 4 0 0 0 0 -2 -2 1 0 0 0 -2 -2 1 0 0 2ζ97+2ζ92 2ζ95+2ζ94 2ζ98+2ζ9 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 0 0 0 0 orthogonal lifted from S3×D9 ρ21 4 4 0 0 0 0 -2 -2 1 0 0 0 -2 -2 1 0 0 2ζ95+2ζ94 2ζ98+2ζ9 2ζ97+2ζ92 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 0 0 0 0 orthogonal lifted from S3×D9 ρ22 4 4 0 0 0 0 -2 -2 1 0 0 0 -2 -2 1 0 0 2ζ98+2ζ9 2ζ97+2ζ92 2ζ95+2ζ94 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 0 0 0 0 orthogonal lifted from S3×D9 ρ23 6 -2 0 0 -2 0 -3 6 -3 2 0 0 1 -2 1 0 1 0 0 0 0 0 0 -1 0 0 0 orthogonal lifted from S3×S4 ρ24 6 -2 6 -2 0 0 6 -3 -3 0 0 -3 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C3.S4 ρ25 6 -2 0 0 2 0 -3 6 -3 -2 0 0 1 -2 1 0 -1 0 0 0 0 0 0 1 0 0 0 orthogonal lifted from S3×S4 ρ26 6 -2 -6 2 0 0 6 -3 -3 0 0 3 -2 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C3.S4 ρ27 12 -4 0 0 0 0 -6 -6 3 0 0 0 2 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of S3×C3.S4
On 36 points
Generators in S36
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 28)(10 27)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)
(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)(2 19)(4 21)(5 22)(7 24)(8 25)(10 29)(11 30)(13 32)(14 33)(16 35)(17 36)
(2 19)(3 20)(5 22)(6 23)(8 25)(9 26)(11 30)(12 31)(14 33)(15 34)(17 36)(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)| (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,28)(10,27)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(11,30)(13,32)(14,33)(16,35)(17,36), (2,19)(3,20)(5,22)(6,23)(8,25)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(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( (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18)(19,25,22)(20,26,23)(21,27,24)(28,31,34)(29,32,35)(30,33,36), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,28)(10,27)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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)(2,19)(4,21)(5,22)(7,24)(8,25)(10,29)(11,30)(13,32)(14,33)(16,35)(17,36), (2,19)(3,20)(5,22)(6,23)(8,25)(9,26)(11,30)(12,31)(14,33)(15,34)(17,36)(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([[(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18),(19,25,22),(20,26,23),(21,27,24),(28,31,34),(29,32,35),(30,33,36)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,28),(10,27),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26)], [(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),(2,19),(4,21),(5,22),(7,24),(8,25),(10,29),(11,30),(13,32),(14,33),(16,35),(17,36)], [(2,19),(3,20),(5,22),(6,23),(8,25),(9,26),(11,30),(12,31),(14,33),(15,34),(17,36),(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 S3×C3.S4 in GL7(𝔽37)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 36 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 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 19 0 0 0 0 0 35 36 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 36 0 0 0 0 0 0 0 36 0 0 0 0 0 1 1 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 36 0 36
,
 6 13 0 0 0 0 0 22 17 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 36 35 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 35 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 36 35 0 0 0 0 0 0 1

G:=sub<GL(7,GF(37))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,35,0,0,0,0,0,19,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,1,0,0,0,0,0,36,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[6,22,0,0,0,0,0,13,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,1,0,0,0,0,0,36,0,0,0,0,0,0,35,0,1],[1,35,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,36,0,0,0,0,0,0,35,1] >;

S3×C3.S4 in GAP, Magma, Sage, TeX

S_3\times C_3.S_4
% in TeX

G:=Group("S3xC3.S4");
// GroupNames label

G:=SmallGroup(432,522);
// by ID

G=gap.SmallGroup(432,522);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,471,394,675,1271,4548,2287,2659,3989]);
// Polycyclic

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

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