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## G = S3×SL2(𝔽3)  order 144 = 24·32

### Direct product of S3 and SL2(𝔽3)

Aliases: S3×SL2(𝔽3), D6.A4, (S3×Q8)⋊C3, (C3×Q8)⋊C6, C2.3(S3×A4), C6.2(C2×A4), Q82(C3×S3), C3⋊(C2×SL2(𝔽3)), (C3×SL2(𝔽3))⋊3C2, SL2(ℤ/6ℤ), SmallGroup(144,128)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — S3×SL2(𝔽3)
 Chief series C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3)
 Lower central C3×Q8 — S3×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for S3×SL2(𝔽3)
G = < a,b,c,d,e | a3=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Character table of S3×SL2(𝔽3)

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 12 size 1 1 3 3 2 4 4 8 8 6 18 2 4 4 8 8 12 12 12 12 12 ρ1 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 linear of order 2 ρ3 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 linear of order 3 ρ4 1 1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 linear of order 6 ρ5 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 linear of order 3 ρ6 1 1 -1 -1 1 ζ3 ζ32 ζ32 ζ3 1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 linear of order 6 ρ7 2 2 0 0 -1 2 2 -1 -1 2 0 -1 2 2 -1 -1 0 0 0 0 -1 orthogonal lifted from S3 ρ8 2 -2 2 -2 2 -1 -1 -1 -1 0 0 -2 1 1 1 1 -1 -1 1 1 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 -2 2 2 -1 -1 -1 -1 0 0 -2 1 1 1 1 1 1 -1 -1 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ10 2 -2 2 -2 2 ζ65 ζ6 ζ6 ζ65 0 0 -2 ζ3 ζ32 ζ32 ζ3 ζ65 ζ6 ζ3 ζ32 0 complex lifted from SL2(𝔽3) ρ11 2 2 0 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 0 0 -1 complex lifted from C3×S3 ρ12 2 -2 2 -2 2 ζ6 ζ65 ζ65 ζ6 0 0 -2 ζ32 ζ3 ζ3 ζ32 ζ6 ζ65 ζ32 ζ3 0 complex lifted from SL2(𝔽3) ρ13 2 2 0 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 0 0 -1 complex lifted from C3×S3 ρ14 2 -2 -2 2 2 ζ6 ζ65 ζ65 ζ6 0 0 -2 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 0 complex lifted from SL2(𝔽3) ρ15 2 -2 -2 2 2 ζ65 ζ6 ζ6 ζ65 0 0 -2 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 0 complex lifted from SL2(𝔽3) ρ16 3 3 -3 -3 3 0 0 0 0 -1 1 3 0 0 0 0 0 0 0 0 -1 orthogonal lifted from C2×A4 ρ17 3 3 3 3 3 0 0 0 0 -1 -1 3 0 0 0 0 0 0 0 0 -1 orthogonal lifted from A4 ρ18 4 -4 0 0 -2 -2 -2 1 1 0 0 2 2 2 -1 -1 0 0 0 0 0 symplectic faithful, Schur index 2 ρ19 4 -4 0 0 -2 1+√-3 1-√-3 ζ3 ζ32 0 0 2 -1-√-3 -1+√-3 ζ65 ζ6 0 0 0 0 0 complex faithful ρ20 4 -4 0 0 -2 1-√-3 1+√-3 ζ32 ζ3 0 0 2 -1+√-3 -1-√-3 ζ6 ζ65 0 0 0 0 0 complex faithful ρ21 6 6 0 0 -3 0 0 0 0 -2 0 -3 0 0 0 0 0 0 0 0 1 orthogonal lifted from S3×A4

Permutation representations of S3×SL2(𝔽3)
On 24 points - transitive group 24T249
Generators in S24
(1 9 16)(2 10 13)(3 11 14)(4 12 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(1 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 14)(10 15)(11 16)(12 13)(21 23)(22 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 24 3 22)(2 23 4 21)(5 13 7 15)(6 16 8 14)(9 18 11 20)(10 17 12 19)
(2 23 24)(4 21 22)(5 6 15)(7 8 13)(10 17 18)(12 19 20)

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

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

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

G:=TransitiveGroup(24,249);

S3×SL2(𝔽3) is a maximal subgroup of   D6.S4  D6.2S4  SL2(𝔽3).11D6  D12.A4  D18.A4  Q8⋊He3⋊C2
S3×SL2(𝔽3) is a maximal quotient of   D18.A4  Q8⋊He3⋊C2

Matrix representation of S3×SL2(𝔽3) in GL4(𝔽5) generated by

 3 2 2 1 4 4 1 1 1 3 3 3 2 3 4 3
,
 1 1 4 2 4 0 2 2 0 1 0 2 3 0 2 4
,
 0 4 1 2 0 3 1 1 1 2 3 3 4 3 4 4
,
 1 3 1 4 1 2 0 4 3 4 0 1 3 3 1 2
,
 3 3 0 0 4 1 0 0 1 1 0 1 2 0 4 4
G:=sub<GL(4,GF(5))| [3,4,1,2,2,4,3,3,2,1,3,4,1,1,3,3],[1,4,0,3,1,0,1,0,4,2,0,2,2,2,2,4],[0,0,1,4,4,3,2,3,1,1,3,4,2,1,3,4],[1,1,3,3,3,2,4,3,1,0,0,1,4,4,1,2],[3,4,1,2,3,1,1,0,0,0,0,4,0,0,1,4] >;

S3×SL2(𝔽3) in GAP, Magma, Sage, TeX

S_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("S3xSL(2,3)");
// GroupNames label

G:=SmallGroup(144,128);
// by ID

G=gap.SmallGroup(144,128);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-2,170,230,81,351,165,1444]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^3=1,d^2=c^2,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,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations

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