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## G = PSO4+ (𝔽3)  order 288 = 25·32

### Projective special orthogonal group of + type on 𝔽34

Aliases: PSO+4(𝔽3), A4⋊S4, A423C2, C24⋊(C3⋊S3), C22⋊A42S3, C221(C3⋊S4), (C22×A4)⋊2S3, Hol(A4), SmallGroup(288,1026)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — A42 — PSO4+ (𝔽3)
 Chief series C1 — C22 — C24 — C22×A4 — A42 — PSO4+ (𝔽3)
 Lower central A42 — PSO4+ (𝔽3)
 Upper central C1

Generators and relations for PSO4+ (𝔽3)
G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f3=g2=1, cac-1=gbg=ab=ba, ad=da, ae=ea, af=fa, ag=ga, cbc-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 916 in 95 conjugacy classes, 12 normal (4 characteristic)
C1, C2 [×4], C3 [×4], C4 [×3], C22 [×2], C22 [×8], S3 [×4], C6 [×2], C2×C4 [×3], D4 [×6], C23 [×4], C32, Dic3 [×2], A4 [×2], A4 [×6], D6 [×2], C2×C6 [×2], C22⋊C4 [×3], C2×D4 [×3], C24, C3⋊S3, C3⋊D4 [×2], S4 [×8], C2×A4 [×2], C22≀C2, C3×A4 [×2], A4⋊C4 [×2], C2×S4 [×2], C22×A4 [×2], C22⋊A4 [×2], C3⋊S4 [×2], A4⋊D4 [×2], C22⋊S4 [×2], A42, PSO4+ (𝔽3)
Quotients: C1, C2, S3 [×4], C3⋊S3, S4 [×2], C3⋊S4 [×2], PSO4+ (𝔽3)

Character table of PSO4+ (𝔽3)

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 6A 6B size 1 3 3 9 36 8 8 32 32 36 36 36 24 24 ρ1 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 linear of order 2 ρ3 2 2 2 2 0 2 -1 -1 -1 0 0 0 -1 2 orthogonal lifted from S3 ρ4 2 2 2 2 0 -1 -1 -1 2 0 0 0 -1 -1 orthogonal lifted from S3 ρ5 2 2 2 2 0 -1 -1 2 -1 0 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 2 0 -1 2 -1 -1 0 0 0 2 -1 orthogonal lifted from S3 ρ7 3 -1 3 -1 1 0 3 0 0 -1 1 -1 -1 0 orthogonal lifted from S4 ρ8 3 3 -1 -1 1 3 0 0 0 -1 -1 1 0 -1 orthogonal lifted from S4 ρ9 3 -1 3 -1 -1 0 3 0 0 1 -1 1 -1 0 orthogonal lifted from S4 ρ10 3 3 -1 -1 -1 3 0 0 0 1 1 -1 0 -1 orthogonal lifted from S4 ρ11 6 -2 6 -2 0 0 -3 0 0 0 0 0 1 0 orthogonal lifted from C3⋊S4 ρ12 6 6 -2 -2 0 -3 0 0 0 0 0 0 0 1 orthogonal lifted from C3⋊S4 ρ13 9 -3 -3 1 1 0 0 0 0 1 -1 -1 0 0 orthogonal faithful ρ14 9 -3 -3 1 -1 0 0 0 0 -1 1 1 0 0 orthogonal faithful

Permutation representations of PSO4+ (𝔽3)
On 12 points - transitive group 12T127
Generators in S12
(1 11)(2 9)(3 6)(4 8)(5 12)(7 10)
(1 4)(2 12)(3 7)(5 9)(6 10)(8 11)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 8)(2 9)(3 7)(4 11)(5 12)(6 10)
(1 11)(2 12)(3 10)(4 8)(5 9)(6 7)
(1 3 2)(4 7 12)(5 8 10)(6 9 11)
(2 3)(4 8)(5 7)(6 9)(10 12)

G:=sub<Sym(12)| (1,11)(2,9)(3,6)(4,8)(5,12)(7,10), (1,4)(2,12)(3,7)(5,9)(6,10)(8,11), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,8)(2,9)(3,7)(4,11)(5,12)(6,10), (1,11)(2,12)(3,10)(4,8)(5,9)(6,7), (1,3,2)(4,7,12)(5,8,10)(6,9,11), (2,3)(4,8)(5,7)(6,9)(10,12)>;

G:=Group( (1,11)(2,9)(3,6)(4,8)(5,12)(7,10), (1,4)(2,12)(3,7)(5,9)(6,10)(8,11), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,8)(2,9)(3,7)(4,11)(5,12)(6,10), (1,11)(2,12)(3,10)(4,8)(5,9)(6,7), (1,3,2)(4,7,12)(5,8,10)(6,9,11), (2,3)(4,8)(5,7)(6,9)(10,12) );

G=PermutationGroup([(1,11),(2,9),(3,6),(4,8),(5,12),(7,10)], [(1,4),(2,12),(3,7),(5,9),(6,10),(8,11)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,8),(2,9),(3,7),(4,11),(5,12),(6,10)], [(1,11),(2,12),(3,10),(4,8),(5,9),(6,7)], [(1,3,2),(4,7,12),(5,8,10),(6,9,11)], [(2,3),(4,8),(5,7),(6,9),(10,12)])

G:=TransitiveGroup(12,127);

On 16 points - transitive group 16T710
Generators in S16
(1 6)(2 10)(3 12)(4 15)(5 7)(8 9)(11 13)(14 16)
(1 7)(2 8)(3 13)(4 16)(5 6)(9 10)(11 12)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 4)(2 3)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)
(1 2)(3 4)(5 9)(6 10)(7 8)(11 14)(12 15)(13 16)
(1 3 2)(5 11 9)(6 12 10)(7 13 8)
(1 2)(5 8)(6 10)(7 9)(11 13)(14 16)

G:=sub<Sym(16)| (1,6)(2,10)(3,12)(4,15)(5,7)(8,9)(11,13)(14,16), (1,7)(2,8)(3,13)(4,16)(5,6)(9,10)(11,12)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,4)(2,3)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (1,2)(3,4)(5,9)(6,10)(7,8)(11,14)(12,15)(13,16), (1,3,2)(5,11,9)(6,12,10)(7,13,8), (1,2)(5,8)(6,10)(7,9)(11,13)(14,16)>;

G:=Group( (1,6)(2,10)(3,12)(4,15)(5,7)(8,9)(11,13)(14,16), (1,7)(2,8)(3,13)(4,16)(5,6)(9,10)(11,12)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,4)(2,3)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12), (1,2)(3,4)(5,9)(6,10)(7,8)(11,14)(12,15)(13,16), (1,3,2)(5,11,9)(6,12,10)(7,13,8), (1,2)(5,8)(6,10)(7,9)(11,13)(14,16) );

G=PermutationGroup([(1,6),(2,10),(3,12),(4,15),(5,7),(8,9),(11,13),(14,16)], [(1,7),(2,8),(3,13),(4,16),(5,6),(9,10),(11,12),(14,15)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)], [(1,4),(2,3),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12)], [(1,2),(3,4),(5,9),(6,10),(7,8),(11,14),(12,15),(13,16)], [(1,3,2),(5,11,9),(6,12,10),(7,13,8)], [(1,2),(5,8),(6,10),(7,9),(11,13),(14,16)])

G:=TransitiveGroup(16,710);

On 18 points - transitive group 18T116
Generators in S18
(2 18)(3 16)(4 7)(5 8)(11 13)(12 14)
(1 17)(3 16)(5 8)(6 9)(10 15)(12 14)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 17)(2 18)(3 16)(4 7)(5 8)(6 9)
(4 7)(5 8)(6 9)(10 15)(11 13)(12 14)
(1 9 10)(2 7 11)(3 8 12)(4 13 18)(5 14 16)(6 15 17)
(1 17)(2 16)(3 18)(4 12)(5 11)(6 10)(7 14)(8 13)(9 15)

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

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

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

G:=TransitiveGroup(18,116);

On 18 points - transitive group 18T117
Generators in S18
(2 18)(3 16)(4 7)(5 8)(11 13)(12 14)
(1 17)(3 16)(5 8)(6 9)(10 15)(12 14)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 17)(2 18)(3 16)(4 7)(5 8)(6 9)
(4 7)(5 8)(6 9)(10 15)(11 13)(12 14)
(1 9 10)(2 7 11)(3 8 12)(4 13 18)(5 14 16)(6 15 17)
(2 3)(4 14)(5 13)(6 15)(7 12)(8 11)(9 10)(16 18)

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

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

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

G:=TransitiveGroup(18,117);

On 24 points - transitive group 24T636
Generators in S24
(2 24)(3 22)(5 9)(6 7)(10 15)(11 13)(17 20)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(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)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 19)(2 20)(3 21)(4 14)(5 15)(6 13)(7 11)(8 12)(9 10)(16 23)(17 24)(18 22)
(4 14 16)(5 15 17)(6 13 18)(7 11 21)(8 12 19)(9 10 20)
(2 3)(4 14)(5 13)(6 15)(7 10)(8 12)(9 11)(17 18)(20 21)(22 24)

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

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

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

G:=TransitiveGroup(24,636);

On 24 points - transitive group 24T637
Generators in S24
(2 24)(3 22)(5 9)(6 7)(10 15)(11 13)(17 20)(18 21)
(1 23)(3 22)(4 8)(6 7)(11 13)(12 14)(16 19)(18 21)
(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)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 19)(2 20)(3 21)(4 14)(5 15)(6 13)(7 11)(8 12)(9 10)(16 23)(17 24)(18 22)
(4 14 16)(5 15 17)(6 13 18)(7 11 21)(8 12 19)(9 10 20)
(1 23)(2 22)(3 24)(4 12)(5 11)(6 10)(7 15)(8 14)(9 13)(16 19)(17 21)(18 20)

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

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

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

G:=TransitiveGroup(24,637);

On 24 points - transitive group 24T693
Generators in S24
(1 18)(2 11)(3 14)(4 19)(5 9)(6 24)(7 21)(8 22)(10 15)(12 17)(13 16)(20 23)
(1 15)(2 16)(3 12)(4 22)(5 20)(6 7)(8 19)(9 23)(10 18)(11 13)(14 17)(21 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 18)(2 16)(3 17)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(19 22)(20 23)(21 24)
(1 10)(2 11)(3 12)(4 22)(5 23)(6 24)(7 21)(8 19)(9 20)(13 16)(14 17)(15 18)
(1 2 3)(4 20 24)(5 21 22)(6 19 23)(7 8 9)(10 13 17)(11 14 18)(12 15 16)
(1 8)(2 7)(3 9)(4 15)(5 14)(6 13)(10 19)(11 21)(12 20)(16 24)(17 23)(18 22)

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

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

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

G:=TransitiveGroup(24,693);

Polynomial with Galois group PSO4+ (𝔽3) over ℚ
actionf(x)Disc(f)
12T127x12-54x9-315x8+4372x6+3996x5-22005x4-13176x3+17484x2-22518x+1775252·316·59·178·892·29692·63112·336307692

Matrix representation of PSO4+ (𝔽3) in GL6(ℤ)

 0 1 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 0 1 0 0 0 -1 -1 -1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 1 -1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 1 0 0 0 -1 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 1 -1 0 0 0 0 0 -1 1
,
 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1 0 0 0 0 -1 0 1

G:=sub<GL(6,Integers())| [0,1,-1,0,0,0,1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,1,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,-1,-1,-1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,0,0,0,0,0,1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,-1,-1,0,0,0,0,1,0,0,0,0,0,0,1] >;

PSO4+ (𝔽3) in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,1026);
// by ID

G=gap.SmallGroup(288,1026);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,57,254,1011,269,634,123,6053,4548,3534,1777]);
// Polycyclic

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

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