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## G = C2×AΓL1(𝔽9)  order 288 = 25·32

### Direct product of C2 and AΓL1(𝔽9)

Aliases: C2×AΓL1(𝔽9), F9⋊C22, PSU3(𝔽2)⋊C22, C3⋊S3⋊SD16, (C3×C6)⋊SD16, (C2×F9)⋊3C2, C32⋊(C2×SD16), S3≀C2.C22, C32⋊C4.3D4, C32⋊C4.C23, (C2×PSU3(𝔽2))⋊2C2, C3⋊S3.(C2×D4), (C2×C3⋊S3).4D4, (C2×S3≀C2).2C2, (C2×C32⋊C4).8C22, SmallGroup(288,1027)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C4 — C2×AΓL1(𝔽9)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — F9 — AΓL1(𝔽9) — C2×AΓL1(𝔽9)
 Lower central C32 — C3⋊S3 — C32⋊C4 — C2×AΓL1(𝔽9)
 Upper central C1 — C2

Generators and relations for C2×AΓL1(𝔽9)
G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, ebe=b-1c, dcd-1=b, ce=ec, ede=d3 >

Subgroups: 604 in 84 conjugacy classes, 24 normal (14 characteristic)
C1, C2, C2 [×4], C3, C4 [×4], C22 [×5], S3 [×4], C6 [×3], C8 [×2], C2×C4 [×2], D4 [×3], Q8 [×3], C23, C32, D6 [×6], C2×C6, C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C22×S3, C2×SD16, C32⋊C4 [×2], C32⋊C4 [×2], S32 [×3], S3×C6, C2×C3⋊S3, F9 [×2], S3≀C2 [×2], S3≀C2, PSU3(𝔽2) [×2], PSU3(𝔽2), C2×C32⋊C4, C2×C32⋊C4, C2×S32, AΓL1(𝔽9) [×4], C2×F9, C2×S3≀C2, C2×PSU3(𝔽2), C2×AΓL1(𝔽9)
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, C2×SD16, AΓL1(𝔽9), C2×AΓL1(𝔽9)

Character table of C2×AΓL1(𝔽9)

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D size 1 1 9 9 12 12 8 18 18 36 36 8 24 24 18 18 18 18 ρ1 1 1 1 1 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 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ6 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ7 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 0 0 2 -2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 2 -2 0 0 2 -2 2 0 0 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 2 0 0 0 0 -2 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ12 2 2 -2 -2 0 0 2 0 0 0 0 2 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ13 2 -2 -2 2 0 0 2 0 0 0 0 -2 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ14 2 2 -2 -2 0 0 2 0 0 0 0 2 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ15 8 -8 0 0 2 -2 -1 0 0 0 0 1 1 -1 0 0 0 0 orthogonal faithful ρ16 8 8 0 0 -2 -2 -1 0 0 0 0 -1 1 1 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ17 8 8 0 0 2 2 -1 0 0 0 0 -1 -1 -1 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ18 8 -8 0 0 -2 2 -1 0 0 0 0 1 -1 1 0 0 0 0 orthogonal faithful

Permutation representations of C2×AΓL1(𝔽9)
On 18 points - transitive group 18T110
Generators in S18
(1 2)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 11)
(1 8 4)(2 17 13)(3 5 10)(6 9 7)(11 12 14)(15 18 16)
(1 9 5)(2 18 14)(3 4 6)(7 10 8)(11 17 16)(12 13 15)
(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)
(3 7)(4 10)(6 8)(11 13)(12 16)(15 17)

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

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

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

G:=TransitiveGroup(18,110);

On 24 points - transitive group 24T681
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 17 21)(2 22 18)(4 24 20)(5 12 16)(6 9 13)(8 11 15)
(1 17 21)(2 18 22)(3 23 19)(5 12 16)(6 13 9)(7 10 14)
(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 3)(5 7)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)

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

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

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

G:=TransitiveGroup(24,681);

On 24 points - transitive group 24T682
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 9 21)(2 10 22)(3 23 11)(5 17 13)(6 18 14)(7 15 19)
(2 10 22)(3 11 23)(4 24 12)(6 18 14)(7 19 15)(8 16 20)
(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 5)(2 8)(4 6)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)

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

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

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

G:=TransitiveGroup(24,682);

Matrix representation of C2×AΓL1(𝔽9) in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,0,0,0,1,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0],[0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,1,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,-1,0,0,0],[-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,-1,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,1,0],[-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0] >;

C2×AΓL1(𝔽9) in GAP, Magma, Sage, TeX

C_2\times {\rm AGammaL}_1({\mathbb F}_9)
% in TeX

G:=Group("C2xAGammaL(1,9)");
// GroupNames label

G:=SmallGroup(288,1027);
// by ID

G=gap.SmallGroup(288,1027);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,365,346,80,4037,4716,1202,201,10982,4717,1595,622]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,e*b*e=b^-1*c,d*c*d^-1=b,c*e=e*c,e*d*e=d^3>;
// generators/relations

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