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## G = A4×SL2(𝔽3)  order 288 = 25·32

### Direct product of A4 and SL2(𝔽3)

Aliases: A4×SL2(𝔽3), C2.1A42, (Q8×A4)⋊C3, (C2×A4).A4, Q8⋊A4⋊C3, Q81(C3×A4), (C22×Q8)⋊C32, C23.3(C3×A4), (C22×SL2(𝔽3))⋊C3, C221(C3×SL2(𝔽3)), SmallGroup(288,859)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22×Q8 — A4×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C22×Q8 — C22×SL2(𝔽3) — A4×SL2(𝔽3)
 Lower central C22×Q8 — A4×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for A4×SL2(𝔽3)
G = < a,b,c,d,e,f | a2=b2=c3=d4=f3=1, e2=d2, cac-1=ab=ba, ad=da, ae=ea, af=fa, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede-1=d-1, fdf-1=e, fef-1=de >

Subgroups: 336 in 59 conjugacy classes, 14 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C2×C4, Q8, Q8, C23, C32, C12, A4, A4, C2×C6, C22×C4, C2×Q8, C3×C6, SL2(𝔽3), SL2(𝔽3), C3×Q8, C2×A4, C2×A4, C22×C6, C22×Q8, C3×A4, C4×A4, C2×SL2(𝔽3), C3×SL2(𝔽3), C6×A4, C22×SL2(𝔽3), Q8×A4, Q8⋊A4, A4×SL2(𝔽3)
Quotients: C1, C3, C32, A4, SL2(𝔽3), C3×A4, C3×SL2(𝔽3), A42, A4×SL2(𝔽3)

Character table of A4×SL2(𝔽3)

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 12A 12B size 1 1 3 3 4 4 4 4 16 16 16 16 6 18 4 4 4 4 12 12 12 12 16 16 16 16 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 1 trivial ρ2 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 linear of order 3 ρ5 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 1 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 linear of order 3 ρ8 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 1 ζ32 ζ3 linear of order 3 ρ9 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 linear of order 3 ρ10 2 -2 2 -2 2 2 -1 -1 -1 -1 -1 -1 0 0 -2 -2 1 1 -1 1 1 -1 1 1 1 1 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ11 2 -2 2 -2 -1+√-3 -1-√-3 ζ65 ζ6 ζ65 -1 ζ6 -1 0 0 1-√-3 1+√-3 ζ32 ζ3 ζ65 ζ32 ζ3 ζ6 ζ3 1 1 ζ32 0 0 complex lifted from C3×SL2(𝔽3) ρ12 2 -2 2 -2 -1-√-3 -1+√-3 -1 -1 ζ65 ζ65 ζ6 ζ6 0 0 1+√-3 1-√-3 1 1 -1 1 1 -1 ζ3 ζ3 ζ32 ζ32 0 0 complex lifted from C3×SL2(𝔽3) ρ13 2 -2 2 -2 2 2 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 0 0 -2 -2 ζ3 ζ32 ζ6 ζ3 ζ32 ζ65 ζ3 ζ32 ζ3 ζ32 0 0 complex lifted from SL2(𝔽3) ρ14 2 -2 2 -2 -1-√-3 -1+√-3 ζ6 ζ65 ζ6 -1 ζ65 -1 0 0 1+√-3 1-√-3 ζ3 ζ32 ζ6 ζ3 ζ32 ζ65 ζ32 1 1 ζ3 0 0 complex lifted from C3×SL2(𝔽3) ρ15 2 -2 2 -2 2 2 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 0 0 -2 -2 ζ32 ζ3 ζ65 ζ32 ζ3 ζ6 ζ32 ζ3 ζ32 ζ3 0 0 complex lifted from SL2(𝔽3) ρ16 2 -2 2 -2 -1+√-3 -1-√-3 -1 -1 ζ6 ζ6 ζ65 ζ65 0 0 1-√-3 1+√-3 1 1 -1 1 1 -1 ζ32 ζ32 ζ3 ζ3 0 0 complex lifted from C3×SL2(𝔽3) ρ17 2 -2 2 -2 -1-√-3 -1+√-3 ζ65 ζ6 -1 ζ6 -1 ζ65 0 0 1+√-3 1-√-3 ζ32 ζ3 ζ65 ζ32 ζ3 ζ6 1 ζ32 ζ3 1 0 0 complex lifted from C3×SL2(𝔽3) ρ18 2 -2 2 -2 -1+√-3 -1-√-3 ζ6 ζ65 -1 ζ65 -1 ζ6 0 0 1-√-3 1+√-3 ζ3 ζ32 ζ6 ζ3 ζ32 ζ65 1 ζ3 ζ32 1 0 0 complex lifted from C3×SL2(𝔽3) ρ19 3 3 -1 -1 0 0 3 3 0 0 0 0 3 -1 0 0 3 3 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ20 3 3 3 3 3 3 0 0 0 0 0 0 -1 -1 3 3 0 0 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from A4 ρ21 3 3 -1 -1 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 3 -1 0 0 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 ζ65 ζ6 0 0 0 0 0 0 complex lifted from C3×A4 ρ22 3 3 -1 -1 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 3 -1 0 0 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 ζ6 ζ65 0 0 0 0 0 0 complex lifted from C3×A4 ρ23 3 3 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 0 0 0 0 ζ6 ζ65 complex lifted from C3×A4 ρ24 3 3 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 0 0 0 0 ζ65 ζ6 complex lifted from C3×A4 ρ25 6 -6 -2 2 0 0 -3 -3 0 0 0 0 0 0 0 0 3 3 1 -1 -1 1 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ26 6 -6 -2 2 0 0 3-3√-3/2 3+3√-3/2 0 0 0 0 0 0 0 0 -3-3√-3/2 -3+3√-3/2 ζ3 ζ6 ζ65 ζ32 0 0 0 0 0 0 complex faithful ρ27 6 -6 -2 2 0 0 3+3√-3/2 3-3√-3/2 0 0 0 0 0 0 0 0 -3+3√-3/2 -3-3√-3/2 ζ32 ζ65 ζ6 ζ3 0 0 0 0 0 0 complex faithful ρ28 9 9 -3 -3 0 0 0 0 0 0 0 0 -3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A42

Permutation representations of A4×SL2(𝔽3)
On 24 points - transitive group 24T580
Generators in S24
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 19 11)(2 20 12)(3 17 9)(4 18 10)(5 21 13)(6 22 14)(7 23 15)(8 24 16)
(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 8 3 6)(2 7 4 5)(9 14 11 16)(10 13 12 15)(17 22 19 24)(18 21 20 23)
(2 7 8)(4 5 6)(10 13 14)(12 15 16)(18 21 22)(20 23 24)

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

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

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

G:=TransitiveGroup(24,580);

Matrix representation of A4×SL2(𝔽3) in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 12 12 12 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 12 12 12
,
 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 4 4 4
,
 3 4 0 0 0 4 10 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 4 10 0 0 0 10 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 10 9 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 0 3

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[9,0,0,0,0,0,9,0,0,0,0,0,9,0,4,0,0,0,0,4,0,0,0,9,4],[3,4,0,0,0,4,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,10,0,0,0,10,9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,10,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3] >;

A4×SL2(𝔽3) in GAP, Magma, Sage, TeX

A_4\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("A4xSL(2,3)");
// GroupNames label

G:=SmallGroup(288,859);
// by ID

G=gap.SmallGroup(288,859);
# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,198,94,3784,172,1517,285,124]);
// Polycyclic

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

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