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## G = S3×GL2(𝔽3)  order 288 = 25·32

### Direct product of S3 and GL2(𝔽3)

Aliases: S3×GL2(𝔽3), D6.4S4, SL2(𝔽3)⋊2D6, Q8⋊S32, (C3×Q8)⋊D6, C6.8(C2×S4), (S3×Q8)⋊3S3, C2.11(S3×S4), C6.6S44C2, C31(C2×GL2(𝔽3)), (S3×SL2(𝔽3))⋊3C2, (C3×GL2(𝔽3))⋊4C2, (C3×SL2(𝔽3))⋊2C22, GL2(ℤ/6ℤ), SmallGroup(288,851)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3) — S3×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) — S3×GL2(𝔽3)
 Lower central C3×SL2(𝔽3) — S3×GL2(𝔽3)
 Upper central C1 — C2

Generators and relations for S3×GL2(𝔽3)
G = < a,b,c,d,e,f | a3=b2=c4=e3=f2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >

Subgroups: 758 in 109 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×2], C22 [×5], S3 [×2], S3 [×7], C6, C6 [×5], C8 [×2], C2×C4, D4 [×3], Q8, Q8, C23, C32, Dic3, C12, D6, D6 [×10], C2×C6 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C3⋊C8, C24, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3 [×2], C2×SD16, S32 [×4], S3×C6 [×2], C2×C3⋊S3, S3×C8, C24⋊C2, D4.S3, Q82S3, C3×SD16, GL2(𝔽3), GL2(𝔽3) [×2], C2×SL2(𝔽3), S3×D4, S3×Q8, C3×SL2(𝔽3), C2×S32, S3×SD16, C2×GL2(𝔽3), C3×GL2(𝔽3), C6.6S4, S3×SL2(𝔽3), S3×GL2(𝔽3)
Quotients: C1, C2 [×3], C22, S3 [×2], D6 [×2], S4, S32, GL2(𝔽3) [×2], C2×S4, C2×GL2(𝔽3), S3×S4, S3×GL2(𝔽3)

Character table of S3×GL2(𝔽3)

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 12 24A 24B size 1 1 3 3 12 36 2 8 16 6 18 2 8 16 24 24 24 6 6 18 18 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 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 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 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 linear of order 2 ρ5 2 2 -2 -2 0 0 2 -1 -1 2 -2 2 -1 -1 0 1 1 0 0 0 0 2 0 0 orthogonal lifted from D6 ρ6 2 2 2 2 0 0 2 -1 -1 2 2 2 -1 -1 0 -1 -1 0 0 0 0 2 0 0 orthogonal lifted from S3 ρ7 2 2 0 0 2 0 -1 2 -1 2 0 -1 2 -1 -1 0 0 2 2 0 0 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 0 0 -2 0 -1 2 -1 2 0 -1 2 -1 1 0 0 -2 -2 0 0 -1 1 1 orthogonal lifted from D6 ρ9 2 -2 2 -2 0 0 2 -1 -1 0 0 -2 1 1 0 1 -1 -√-2 √-2 -√-2 √-2 0 √-2 -√-2 complex lifted from GL2(𝔽3) ρ10 2 -2 2 -2 0 0 2 -1 -1 0 0 -2 1 1 0 1 -1 √-2 -√-2 √-2 -√-2 0 -√-2 √-2 complex lifted from GL2(𝔽3) ρ11 2 -2 -2 2 0 0 2 -1 -1 0 0 -2 1 1 0 -1 1 -√-2 √-2 √-2 -√-2 0 √-2 -√-2 complex lifted from GL2(𝔽3) ρ12 2 -2 -2 2 0 0 2 -1 -1 0 0 -2 1 1 0 -1 1 √-2 -√-2 -√-2 √-2 0 -√-2 √-2 complex lifted from GL2(𝔽3) ρ13 3 3 3 3 1 1 3 0 0 -1 -1 3 0 0 1 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ14 3 3 -3 -3 -1 1 3 0 0 -1 1 3 0 0 -1 0 0 1 1 -1 -1 -1 1 1 orthogonal lifted from C2×S4 ρ15 3 3 3 3 -1 -1 3 0 0 -1 -1 3 0 0 -1 0 0 1 1 1 1 -1 1 1 orthogonal lifted from S4 ρ16 3 3 -3 -3 1 -1 3 0 0 -1 1 3 0 0 1 0 0 -1 -1 1 1 -1 -1 -1 orthogonal lifted from C2×S4 ρ17 4 -4 4 -4 0 0 4 1 1 0 0 -4 -1 -1 0 -1 1 0 0 0 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ18 4 -4 -4 4 0 0 4 1 1 0 0 -4 -1 -1 0 1 -1 0 0 0 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ19 4 4 0 0 0 0 -2 -2 1 4 0 -2 -2 1 0 0 0 0 0 0 0 -2 0 0 orthogonal lifted from S32 ρ20 4 -4 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 -2√-2 2√-2 0 0 0 -√-2 √-2 complex faithful ρ21 4 -4 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 2√-2 -2√-2 0 0 0 √-2 -√-2 complex faithful ρ22 6 6 0 0 2 0 -3 0 0 -2 0 -3 0 0 -1 0 0 -2 -2 0 0 1 1 1 orthogonal lifted from S3×S4 ρ23 6 6 0 0 -2 0 -3 0 0 -2 0 -3 0 0 1 0 0 2 2 0 0 1 -1 -1 orthogonal lifted from S3×S4 ρ24 8 -8 0 0 0 0 -4 2 -1 0 0 4 -2 1 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×GL2(𝔽3)
On 24 points - transitive group 24T678
Generators in S24
(1 16 24)(2 13 21)(3 14 22)(4 15 23)(5 11 19)(6 12 20)(7 9 17)(8 10 18)
(1 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(13 23)(14 24)(15 21)(16 22)
(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 11 3 9)(2 10 4 12)(5 22 7 24)(6 21 8 23)(13 18 15 20)(14 17 16 19)
(2 11 10)(4 9 12)(5 8 21)(6 23 7)(13 19 18)(15 17 20)
(1 3)(2 11)(4 9)(5 21)(7 23)(13 19)(14 16)(15 17)(22 24)

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

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

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

G:=TransitiveGroup(24,678);

Matrix representation of S3×GL2(𝔽3) in GL4(𝔽73) generated by

 71 69 0 0 19 1 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 19 1 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 45 56 0 0 29 28
,
 1 0 0 0 0 1 0 0 0 0 57 44 0 0 29 16
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 72 0 0 0 0 72 0 0 0 0 1 72 0 0 0 72
G:=sub<GL(4,GF(73))| [71,19,0,0,69,1,0,0,0,0,1,0,0,0,0,1],[72,19,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,45,29,0,0,56,28],[1,0,0,0,0,1,0,0,0,0,57,29,0,0,44,16],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,72,72] >;

S3×GL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("S3xGL(2,3)");
// GroupNames label

G:=SmallGroup(288,851);
// by ID

G=gap.SmallGroup(288,851);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,93,675,1271,1908,172,768,1153,285,124]);
// Polycyclic

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

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