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## G = S3×F9order 432 = 24·33

### Direct product of S3 and F9

Aliases: S3×F9, C33⋊(C2×C8), C3⋊F93C2, C31(C2×F9), (S3×C32)⋊C8, C33⋊C2⋊C8, (C3×F9)⋊2C2, C323(S3×C8), C33⋊C4.C4, C32⋊C4.4D6, (S3×C3⋊S3).C4, C3⋊S3.1(C4×S3), (S3×C32⋊C4).2C2, (C3×C32⋊C4).5C22, (C3×C3⋊S3).(C2×C4), SmallGroup(432,736)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×F9
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — S3×C32⋊C4 — S3×F9
 Lower central C33 — S3×F9
 Upper central C1

Generators and relations for S3×F9
G = < a,b,c,d,e | a3=b2=c3=d3=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 568 in 58 conjugacy classes, 18 normal (all characteristic)
C1, C2, C3, C3, C4, C22, S3, S3, C6, C8, C2×C4, C32, C32, Dic3, C12, D6, C2×C8, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C33, C32⋊C4, C32⋊C4, S32, C2×C3⋊S3, S3×C8, S3×C32, C3×C3⋊S3, C33⋊C2, F9, F9, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, S3×C3⋊S3, C2×F9, C3×F9, C3⋊F9, S3×C32⋊C4, S3×F9
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D6, C2×C8, C4×S3, S3×C8, F9, C2×F9, S3×F9

Character table of S3×F9

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 24A 24B 24C 24D size 1 3 9 27 2 8 16 9 9 27 27 18 24 9 9 9 9 27 27 27 27 18 18 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 -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 1 -1 1 -1 1 1 1 -1 -1 1 1 1 -1 i -i -i i -i i -i i -1 -1 i -i -i i linear of order 4 ρ6 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 i -i -i i i -i i -i -1 -1 i -i -i i linear of order 4 ρ7 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -i i i -i -i i -i i -1 -1 -i i i -i linear of order 4 ρ8 1 -1 1 -1 1 1 1 -1 -1 1 1 1 -1 -i i i -i i -i i -i -1 -1 -i i i -i linear of order 4 ρ9 1 -1 -1 1 1 1 1 i -i i -i -1 -1 ζ83 ζ85 ζ8 ζ87 ζ87 ζ85 ζ83 ζ8 i -i ζ83 ζ85 ζ8 ζ87 linear of order 8 ρ10 1 -1 -1 1 1 1 1 -i i -i i -1 -1 ζ8 ζ87 ζ83 ζ85 ζ85 ζ87 ζ8 ζ83 -i i ζ8 ζ87 ζ83 ζ85 linear of order 8 ρ11 1 1 -1 -1 1 1 1 i -i -i i -1 1 ζ83 ζ85 ζ8 ζ87 ζ83 ζ8 ζ87 ζ85 i -i ζ83 ζ85 ζ8 ζ87 linear of order 8 ρ12 1 1 -1 -1 1 1 1 -i i i -i -1 1 ζ8 ζ87 ζ83 ζ85 ζ8 ζ83 ζ85 ζ87 -i i ζ8 ζ87 ζ83 ζ85 linear of order 8 ρ13 1 -1 -1 1 1 1 1 -i i -i i -1 -1 ζ85 ζ83 ζ87 ζ8 ζ8 ζ83 ζ85 ζ87 -i i ζ85 ζ83 ζ87 ζ8 linear of order 8 ρ14 1 -1 -1 1 1 1 1 i -i i -i -1 -1 ζ87 ζ8 ζ85 ζ83 ζ83 ζ8 ζ87 ζ85 i -i ζ87 ζ8 ζ85 ζ83 linear of order 8 ρ15 1 1 -1 -1 1 1 1 -i i i -i -1 1 ζ85 ζ83 ζ87 ζ8 ζ85 ζ87 ζ8 ζ83 -i i ζ85 ζ83 ζ87 ζ8 linear of order 8 ρ16 1 1 -1 -1 1 1 1 i -i -i i -1 1 ζ87 ζ8 ζ85 ζ83 ζ87 ζ85 ζ83 ζ8 i -i ζ87 ζ8 ζ85 ζ83 linear of order 8 ρ17 2 0 2 0 -1 2 -1 2 2 0 0 -1 0 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ18 2 0 2 0 -1 2 -1 2 2 0 0 -1 0 -2 -2 -2 -2 0 0 0 0 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ19 2 0 2 0 -1 2 -1 -2 -2 0 0 -1 0 -2i 2i 2i -2i 0 0 0 0 1 1 i -i -i i complex lifted from C4×S3 ρ20 2 0 2 0 -1 2 -1 -2 -2 0 0 -1 0 2i -2i -2i 2i 0 0 0 0 1 1 -i i i -i complex lifted from C4×S3 ρ21 2 0 -2 0 -1 2 -1 2i -2i 0 0 1 0 2ζ83 2ζ85 2ζ8 2ζ87 0 0 0 0 -i i ζ87 ζ8 ζ85 ζ83 complex lifted from S3×C8 ρ22 2 0 -2 0 -1 2 -1 -2i 2i 0 0 1 0 2ζ8 2ζ87 2ζ83 2ζ85 0 0 0 0 i -i ζ85 ζ83 ζ87 ζ8 complex lifted from S3×C8 ρ23 2 0 -2 0 -1 2 -1 2i -2i 0 0 1 0 2ζ87 2ζ8 2ζ85 2ζ83 0 0 0 0 -i i ζ83 ζ85 ζ8 ζ87 complex lifted from S3×C8 ρ24 2 0 -2 0 -1 2 -1 -2i 2i 0 0 1 0 2ζ85 2ζ83 2ζ87 2ζ8 0 0 0 0 i -i ζ8 ζ87 ζ83 ζ85 complex lifted from S3×C8 ρ25 8 8 0 0 8 -1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F9 ρ26 8 -8 0 0 8 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×F9 ρ27 16 0 0 0 -8 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×F9
On 24 points - transitive group 24T1336
Generators in S24
(1 14 22)(2 15 23)(3 16 24)(4 9 17)(5 10 18)(6 11 19)(7 12 20)(8 13 21)
(1 5)(2 6)(3 7)(4 8)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(2 15 23)(3 16 24)(4 17 9)(6 19 11)(7 20 12)(8 13 21)
(1 14 22)(3 16 24)(4 9 17)(5 18 10)(7 20 12)(8 21 13)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,1336);

On 27 points - transitive group 27T138
Generators in S27
(1 2 3)(4 20 13)(5 21 14)(6 22 15)(7 23 16)(8 24 17)(9 25 18)(10 26 19)(11 27 12)
(2 3)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)
(1 13 17)(2 4 8)(3 20 24)(5 11 10)(6 7 9)(12 19 14)(15 16 18)(21 27 26)(22 23 25)
(1 14 18)(2 5 9)(3 21 25)(4 11 6)(7 8 10)(12 15 13)(16 17 19)(20 27 22)(23 24 26)
(4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27)

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

G:=Group( (1,2,3)(4,20,13)(5,21,14)(6,22,15)(7,23,16)(8,24,17)(9,25,18)(10,26,19)(11,27,12), (2,3)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27), (1,13,17)(2,4,8)(3,20,24)(5,11,10)(6,7,9)(12,19,14)(15,16,18)(21,27,26)(22,23,25), (1,14,18)(2,5,9)(3,21,25)(4,11,6)(7,8,10)(12,15,13)(16,17,19)(20,27,22)(23,24,26), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27) );

G=PermutationGroup([[(1,2,3),(4,20,13),(5,21,14),(6,22,15),(7,23,16),(8,24,17),(9,25,18),(10,26,19),(11,27,12)], [(2,3),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27)], [(1,13,17),(2,4,8),(3,20,24),(5,11,10),(6,7,9),(12,19,14),(15,16,18),(21,27,26),(22,23,25)], [(1,14,18),(2,5,9),(3,21,25),(4,11,6),(7,8,10),(12,15,13),(16,17,19),(20,27,22),(23,24,26)], [(4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27)]])

G:=TransitiveGroup(27,138);

Matrix representation of S3×F9 in GL10(ℤ)

 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 -1 1 0 0
,
 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

G:=sub<GL(10,Integers())| [-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

S3×F9 in GAP, Magma, Sage, TeX

S_3\times F_9
% in TeX

G:=Group("S3xF9");
// GroupNames label

G:=SmallGroup(432,736);
// by ID

G=gap.SmallGroup(432,736);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,36,58,1131,718,165,348,691,530,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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