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G = A4×SL2(𝔽3)  order 288 = 25·32

Direct product of A4 and SL2(𝔽3)

direct product, non-abelian, soluble

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

C1C2C22×Q8 — A4×SL2(𝔽3)
C1C2Q8C22×Q8C22×SL2(𝔽3) — A4×SL2(𝔽3)
C22×Q8 — A4×SL2(𝔽3)
C1C2

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 [×2], C3 [×4], C4 [×2], C22, C22 [×2], C6 [×6], C2×C4 [×2], Q8, Q8 [×3], C23, C32, C12, A4, A4 [×2], C2×C6 [×3], C22×C4, C2×Q8 [×2], C3×C6, SL2(𝔽3), SL2(𝔽3) [×4], C3×Q8, C2×A4, C2×A4 [×2], C22×C6, C22×Q8, C3×A4, C4×A4, C2×SL2(𝔽3), C3×SL2(𝔽3), C6×A4, C22×SL2(𝔽3), Q8×A4, Q8⋊A4 [×2], A4×SL2(𝔽3)
Quotients: C1, C3 [×4], C32, A4 [×2], SL2(𝔽3), C3×A4 [×2], C3×SL2(𝔽3), A42, A4×SL2(𝔽3)

Character table of A4×SL2(𝔽3)

 class 12A2B2C3A3B3C3D3E3F3G3H4A4B6A6B6C6D6E6F6G6H6I6J6K6L12A12B
 size 1133444416161616618444412121212161616162424
ρ11111111111111111111111111111    trivial
ρ21111ζ3ζ3211ζ32ζ32ζ3ζ311ζ3ζ32111111ζ32ζ32ζ3ζ3ζ3ζ32    linear of order 3
ρ31111ζ32ζ3ζ32ζ3ζ321ζ3111ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3211ζ3ζ32ζ3    linear of order 3
ρ4111111ζ3ζ32ζ32ζ3ζ3ζ321111ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ311    linear of order 3
ρ51111ζ32ζ311ζ3ζ3ζ32ζ3211ζ32ζ3111111ζ3ζ3ζ32ζ32ζ32ζ3    linear of order 3
ρ61111ζ3ζ32ζ32ζ31ζ31ζ3211ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ31ζ3ζ321ζ3ζ32    linear of order 3
ρ71111ζ3ζ32ζ3ζ32ζ31ζ32111ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ311ζ32ζ3ζ32    linear of order 3
ρ81111ζ32ζ3ζ3ζ321ζ321ζ311ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ321ζ32ζ31ζ32ζ3    linear of order 3
ρ9111111ζ32ζ3ζ3ζ32ζ32ζ31111ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ3211    linear of order 3
ρ102-22-222-1-1-1-1-1-100-2-211-111-1111100    symplectic lifted from SL2(𝔽3), Schur index 2
ρ112-22-2-1+-3-1--3ζ65ζ6ζ65-1ζ6-1001--31+-3ζ32ζ3ζ65ζ32ζ3ζ6ζ311ζ3200    complex lifted from C3×SL2(𝔽3)
ρ122-22-2-1--3-1+-3-1-1ζ65ζ65ζ6ζ6001+-31--311-111-1ζ3ζ3ζ32ζ3200    complex lifted from C3×SL2(𝔽3)
ρ132-22-222ζ6ζ65ζ65ζ6ζ6ζ6500-2-2ζ3ζ32ζ6ζ3ζ32ζ65ζ3ζ32ζ3ζ3200    complex lifted from SL2(𝔽3)
ρ142-22-2-1--3-1+-3ζ6ζ65ζ6-1ζ65-1001+-31--3ζ3ζ32ζ6ζ3ζ32ζ65ζ3211ζ300    complex lifted from C3×SL2(𝔽3)
ρ152-22-222ζ65ζ6ζ6ζ65ζ65ζ600-2-2ζ32ζ3ζ65ζ32ζ3ζ6ζ32ζ3ζ32ζ300    complex lifted from SL2(𝔽3)
ρ162-22-2-1+-3-1--3-1-1ζ6ζ6ζ65ζ65001--31+-311-111-1ζ32ζ32ζ3ζ300    complex lifted from C3×SL2(𝔽3)
ρ172-22-2-1--3-1+-3ζ65ζ6-1ζ6-1ζ65001+-31--3ζ32ζ3ζ65ζ32ζ3ζ61ζ32ζ3100    complex lifted from C3×SL2(𝔽3)
ρ182-22-2-1+-3-1--3ζ6ζ65-1ζ65-1ζ6001--31+-3ζ3ζ32ζ6ζ3ζ32ζ651ζ3ζ32100    complex lifted from C3×SL2(𝔽3)
ρ1933-1-1003300003-10033-1-1-1-1000000    orthogonal lifted from A4
ρ20333333000000-1-1330000000000-1-1    orthogonal lifted from A4
ρ2133-1-100-3+3-3/2-3-3-3/200003-100-3-3-3/2-3+3-3/2ζ65ζ6ζ65ζ6000000    complex lifted from C3×A4
ρ2233-1-100-3-3-3/2-3+3-3/200003-100-3+3-3/2-3-3-3/2ζ6ζ65ζ6ζ65000000    complex lifted from C3×A4
ρ233333-3-3-3/2-3+3-3/2000000-1-1-3-3-3/2-3+3-3/20000000000ζ6ζ65    complex lifted from C3×A4
ρ243333-3+3-3/2-3-3-3/2000000-1-1-3+3-3/2-3-3-3/20000000000ζ65ζ6    complex lifted from C3×A4
ρ256-6-2200-3-300000000331-1-11000000    symplectic faithful, Schur index 2
ρ266-6-22003-3-3/23+3-3/200000000-3-3-3/2-3+3-3/2ζ3ζ6ζ65ζ32000000    complex faithful
ρ276-6-22003+3-3/23-3-3/200000000-3+3-3/2-3-3-3/2ζ32ζ65ζ6ζ3000000    complex faithful
ρ2899-3-300000000-3100000000000000    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)

10000
01000
00001
00121212
00100
,
10000
01000
00010
00100
00121212
,
90000
09000
00900
00009
00444
,
34000
410000
00100
00010
00001
,
410000
109000
00100
00010
00001
,
10000
109000
00300
00030
00003

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

Export

Character table of A4×SL2(𝔽3) in TeX

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