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G = Ω4+ (𝔽3)  order 288 = 25·32

Omega group of + type on 𝔽34

non-abelian, soluble

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 >

C1
C2
18C2
4C3
4C3
16C3
16C3
3C4
3C4
9C22
12C22
12C22
4C6
4C6
16C6
16C6
16C32
Q8
Q8
3C23
3C23
9D4
9C2×C4
9D4
12A4
12C12
12A4
12C12
16C3×C6
3C4○D4
3C4○D4
9C2×D4
SL2(𝔽3)
SL2(𝔽3)
4SL2(𝔽3)
4SL2(𝔽3)
4SL2(𝔽3)
4SL2(𝔽3)
4C3×Q8
4C3×Q8
12C2×A4
12C2×A4
2+ 1+4
3C4.A4
3C4.A4
4C3×SL2(𝔽3)
4C3×SL2(𝔽3)
Q8.A4
Q8.A4
C23⋊A4
C23⋊A4
Ω4+ (𝔽3)

Character table of Ω4+ (𝔽3)

 class 12A2B3A3B3C3D3E3F3G3H4A4B6A6B6C6D6E6F6G6H12A12B12C12D
 size 11184444161616166644441616161624242424
ρ11111111111111111111111111    trivial
ρ2111ζ32ζ311ζ3ζ3ζ32ζ321111ζ3ζ32ζ3ζ32ζ32ζ31ζ32ζ31    linear of order 3
ρ3111ζ3ζ32ζ32ζ31ζ31ζ3211ζ32ζ3ζ32ζ3ζ31ζ321ζ32ζ3ζ32ζ3    linear of order 3
ρ4111ζ3ζ3211ζ32ζ32ζ3ζ31111ζ32ζ3ζ32ζ3ζ3ζ321ζ3ζ321    linear of order 3
ρ5111ζ32ζ3ζ32ζ3ζ321ζ3111ζ32ζ3ζ3ζ321ζ31ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ6111ζ32ζ3ζ3ζ321ζ321ζ311ζ3ζ32ζ3ζ32ζ321ζ31ζ3ζ32ζ3ζ32    linear of order 3
ρ711111ζ3ζ32ζ32ζ3ζ3ζ3211ζ3ζ3211ζ3ζ3ζ32ζ32ζ311ζ32    linear of order 3
ρ8111ζ3ζ32ζ3ζ32ζ31ζ32111ζ3ζ32ζ32ζ31ζ321ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ911111ζ32ζ3ζ3ζ32ζ32ζ311ζ32ζ311ζ32ζ32ζ3ζ3ζ3211ζ3    linear of order 3
ρ1033-1330000003-1003300000-1-10    orthogonal lifted from A4
ρ1133-100330000-1333000000-100-1    orthogonal lifted from A4
ρ1233-1-3-3√-3/2-3+3√-3/20000003-100-3+3√-3/2-3-3√-3/200000ζ6ζ650    complex lifted from C3×A4
ρ1333-1-3+3√-3/2-3-3√-3/20000003-100-3-3√-3/2-3+3√-3/200000ζ65ζ60    complex lifted from C3×A4
ρ1433-100-3+3√-3/2-3-3√-3/20000-13-3+3√-3/2-3-3√-3/2000000ζ6500ζ6    complex lifted from C3×A4
ρ1533-100-3-3√-3/2-3+3√-3/20000-13-3-3√-3/2-3+3√-3/2000000ζ600ζ65    complex lifted from C3×A4
ρ164-40-2-2-2-21111002222-1-1-1-10000    orthogonal faithful
ρ174-401+√-31-√-31+√-31-√-3ζ321ζ3100-1-√-3-1+√-3-1+√-3-1-√-3-1ζ65-1ζ60000    complex faithful
ρ184-40-2-21-√-31+√-3ζ32ζ3ζ3ζ3200-1+√-3-1-√-322ζ65ζ65ζ6ζ60000    complex faithful
ρ194-401+√-31-√-31-√-31+√-31ζ321ζ300-1+√-3-1-√-3-1+√-3-1-√-3ζ6-1ζ65-10000    complex faithful
ρ204-401-√-31+√-31+√-31-√-31ζ31ζ3200-1-√-3-1+√-3-1-√-3-1+√-3ζ65-1ζ6-10000    complex faithful
ρ214-401+√-31-√-3-2-2ζ3ζ3ζ32ζ320022-1+√-3-1-√-3ζ65ζ6ζ6ζ650000    complex faithful
ρ224-401-√-31+√-31-√-31+√-3ζ31ζ32100-1+√-3-1-√-3-1-√-3-1+√-3-1ζ6-1ζ650000    complex faithful
ρ234-401-√-31+√-3-2-2ζ32ζ32ζ3ζ30022-1-√-3-1+√-3ζ6ζ65ζ65ζ60000    complex faithful
ρ244-40-2-21+√-31-√-3ζ3ζ32ζ32ζ300-1-√-3-1+√-322ζ6ζ6ζ65ζ650000    complex faithful
ρ2599100000000-3-3000000000000    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

000-1
00-10
0100
1000
,
00-10
0001
1000
0-100
,
-1/21/21/21/2
-1/2-1/21/2-1/2
-1/2-1/2-1/21/2
-1/21/2-1/2-1/2
,
0001
0010
0100
1000
,
0100
1000
0001
0010
,
1000
0001
0100
0010
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
 

Export

Subgroup lattice of Ω4+ (𝔽3) in TeX
Character table of Ω4+ (𝔽3) in TeX

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