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G = Ω4+ (𝔽3)  order 288 = 25·32

Omega group of + type on 𝔽34

Aliases: Ω+4(𝔽3), SL2(𝔽3)⋊A4, 2+ 1+4⋊C32, C2.2A42, Q8.A4⋊C3, Q8.(C3×A4), C23⋊A4⋊C3, SmallGroup(288,860)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — 2+ 1+4 — Ω4+ (𝔽3)
 Chief series C1 — C2 — Q8 — 2+ 1+4 — Q8.A4 — Ω4+ (𝔽3)
 Lower central 2+ 1+4 — Ω4+ (𝔽3)
 Upper central C1 — C2

Generators and relations for Ω4+ (𝔽3)
G = < a,b,c,d,e,f | a4=c3=d2=e2=f3=1, b2=a2, bab-1=dad=a-1, cac-1=faf-1=b, ae=ea, cbc-1=fbf-1=ab, bd=db, ebe=a2b, dcd=a-1c, ece=a-1bc, cf=fc, fdf-1=de=ed, fef-1=d >

18C2
4C3
4C3
16C3
16C3
3C4
3C4
9C22
12C22
12C22
4C6
4C6
16C6
16C6
16C32
3C23
3C23
9D4
9D4
12A4
12C12
12A4
12C12
16C3×C6
12C2×A4
12C2×A4

Character table of Ω4+ (𝔽3)

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 3G 3H 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B 12C 12D size 1 1 18 4 4 4 4 16 16 16 16 6 6 4 4 4 4 16 16 16 16 24 24 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 trivial ρ2 1 1 1 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 ζ32 ζ3 1 linear of order 3 ρ3 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 1 1 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 ζ3 ζ32 1 linear of order 3 ρ5 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 1 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ7 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 ζ32 linear of order 3 ρ8 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 1 1 1 ζ3 ζ32 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ9 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 ζ3 linear of order 3 ρ10 3 3 -1 3 3 0 0 0 0 0 0 3 -1 0 0 3 3 0 0 0 0 0 -1 -1 0 orthogonal lifted from A4 ρ11 3 3 -1 0 0 3 3 0 0 0 0 -1 3 3 3 0 0 0 0 0 0 -1 0 0 -1 orthogonal lifted from A4 ρ12 3 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3 -1 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 ζ6 ζ65 0 complex lifted from C3×A4 ρ13 3 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3 -1 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 ζ65 ζ6 0 complex lifted from C3×A4 ρ14 3 3 -1 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 0 0 ζ6 complex lifted from C3×A4 ρ15 3 3 -1 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 0 0 ζ65 complex lifted from C3×A4 ρ16 4 -4 0 -2 -2 -2 -2 1 1 1 1 0 0 2 2 2 2 -1 -1 -1 -1 0 0 0 0 orthogonal faithful ρ17 4 -4 0 1+√-3 1-√-3 1+√-3 1-√-3 ζ32 1 ζ3 1 0 0 -1-√-3 -1+√-3 -1+√-3 -1-√-3 -1 ζ65 -1 ζ6 0 0 0 0 complex faithful ρ18 4 -4 0 -2 -2 1-√-3 1+√-3 ζ32 ζ3 ζ3 ζ32 0 0 -1+√-3 -1-√-3 2 2 ζ65 ζ65 ζ6 ζ6 0 0 0 0 complex faithful ρ19 4 -4 0 1+√-3 1-√-3 1-√-3 1+√-3 1 ζ32 1 ζ3 0 0 -1+√-3 -1-√-3 -1+√-3 -1-√-3 ζ6 -1 ζ65 -1 0 0 0 0 complex faithful ρ20 4 -4 0 1-√-3 1+√-3 1+√-3 1-√-3 1 ζ3 1 ζ32 0 0 -1-√-3 -1+√-3 -1-√-3 -1+√-3 ζ65 -1 ζ6 -1 0 0 0 0 complex faithful ρ21 4 -4 0 1+√-3 1-√-3 -2 -2 ζ3 ζ3 ζ32 ζ32 0 0 2 2 -1+√-3 -1-√-3 ζ65 ζ6 ζ6 ζ65 0 0 0 0 complex faithful ρ22 4 -4 0 1-√-3 1+√-3 1-√-3 1+√-3 ζ3 1 ζ32 1 0 0 -1+√-3 -1-√-3 -1-√-3 -1+√-3 -1 ζ6 -1 ζ65 0 0 0 0 complex faithful ρ23 4 -4 0 1-√-3 1+√-3 -2 -2 ζ32 ζ32 ζ3 ζ3 0 0 2 2 -1-√-3 -1+√-3 ζ6 ζ65 ζ65 ζ6 0 0 0 0 complex faithful ρ24 4 -4 0 -2 -2 1+√-3 1-√-3 ζ3 ζ32 ζ32 ζ3 0 0 -1-√-3 -1+√-3 2 2 ζ6 ζ6 ζ65 ζ65 0 0 0 0 complex faithful ρ25 9 9 1 0 0 0 0 0 0 0 0 -3 -3 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A42

Permutation representations of Ω4+ (𝔽3)
On 24 points - transitive group 24T685
Generators in S24
(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 12 3 10)(2 11 4 9)(5 21 7 23)(6 24 8 22)(13 19 15 17)(14 18 16 20)
(1 23 16)(2 8 20)(3 21 14)(4 6 18)(5 13 11)(7 15 9)(10 22 17)(12 24 19)
(1 3)(5 22)(6 21)(7 24)(8 23)(10 12)(13 14)(15 16)(17 20)(18 19)
(1 12)(2 9)(3 10)(4 11)(5 22)(6 23)(7 24)(8 21)(13 15)(14 16)
(1 16 23)(2 19 5)(3 14 21)(4 17 7)(6 10 15)(8 12 13)(9 18 22)(11 20 24)

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

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

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

G:=TransitiveGroup(24,685);

Matrix representation of Ω4+ (𝔽3) in GL4(ℚ) generated by

 0 0 0 -1 0 0 -1 0 0 1 0 0 1 0 0 0
,
 0 0 -1 0 0 0 0 1 1 0 0 0 0 -1 0 0
,
 -1/2 1/2 1/2 1/2 -1/2 -1/2 1/2 -1/2 -1/2 -1/2 -1/2 1/2 -1/2 1/2 -1/2 -1/2
,
 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0
G:=sub<GL(4,Rationals())| [0,0,0,1,0,0,1,0,0,-1,0,0,-1,0,0,0],[0,0,1,0,0,0,0,-1,-1,0,0,0,0,1,0,0],[-1/2,-1/2,-1/2,-1/2,1/2,-1/2,-1/2,1/2,1/2,1/2,-1/2,-1/2,1/2,-1/2,1/2,-1/2],[0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0] >;

Ω4+ (𝔽3) in GAP, Magma, Sage, TeX

{\rm Omega}_4^+({\mathbb F}_3)
% in TeX

G:=Group("Omega+(4,3)");
// GroupNames label

G:=SmallGroup(288,860);
// by ID

G=gap.SmallGroup(288,860);
# by ID

G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,198,352,94,521,248,3784,172,1517,285,124]);
// Polycyclic

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

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