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## G = C2.AΓL1(𝔽9)  order 288 = 25·32

### 1st central extension by C2 of AΓL1(𝔽9)

Aliases: C2.1AΓL1(𝔽9), S3≀C2⋊C4, C3⋊S3.D8, (C2×F9)⋊1C2, C32⋊C4.1D4, C32⋊(D4⋊C4), (C3×C6).1SD16, C2.PSU3(𝔽2)⋊1C2, (C2×C3⋊S3).1D4, (C2×S3≀C2).1C2, C32⋊C4.1(C2×C4), C3⋊S3.1(C22⋊C4), (C2×C32⋊C4).1C22, SmallGroup(288,841)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C4 — C2.AΓL1(𝔽9)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C2×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=ad3 >

Subgroups: 524 in 66 conjugacy classes, 16 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C8, C2×C4, D4, C23, C32, D6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3⋊S3, C3×C6, C22×S3, D4⋊C4, C32⋊C4, C32⋊C4, S32, S3×C6, C2×C3⋊S3, F9, S3≀C2, S3≀C2, C2×C32⋊C4, C2×C32⋊C4, C2×S32, C2.PSU3(𝔽2), C2×F9, C2×S3≀C2, C2.AΓL1(𝔽9)
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, D4⋊C4, 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 -i i -1 -1 1 -i -i i i linear of order 4 ρ6 1 -1 -1 1 1 -1 1 -1 1 i -i -1 -1 1 i i -i -i linear of order 4 ρ7 1 -1 -1 1 -1 1 1 -1 1 i -i -1 1 -1 -i -i i i linear of order 4 ρ8 1 -1 -1 1 -1 1 1 -1 1 -i i -1 1 -1 i i -i -i linear of order 4 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 orthogonal lifted from D8 ρ12 2 -2 2 -2 0 0 2 0 0 0 0 -2 0 0 -√2 √2 √2 -√2 orthogonal lifted from D8 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 24 points - transitive group 24T680
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)
(1 11 15)(2 12 16)(3 9 13)(5 23 19)(6 20 24)(8 22 18)
(2 12 16)(3 13 9)(4 10 14)(5 23 19)(6 24 20)(7 21 17)
(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 7)(4 5)(10 23)(11 15)(12 21)(14 19)(16 17)(18 22)

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

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

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

G:=TransitiveGroup(24,680);

On 24 points - transitive group 24T683
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(1 15 21)(3 9 23)(4 10 24)(5 17 11)(7 19 13)(8 20 14)
(1 21 15)(2 16 22)(4 10 24)(5 11 17)(6 18 12)(8 20 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)
(2 8)(3 7)(4 6)(9 13)(10 18)(12 24)(14 22)(16 20)(19 23)

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

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

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

G:=TransitiveGroup(24,683);

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 1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 -1 -1 0 0 0 0 0 0 1 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 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 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,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0] >;

C2.AΓL1(𝔽9) in GAP, Magma, Sage, TeX

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

G:=Group("C2.AGammaL(1,9)");
// GroupNames label

G:=SmallGroup(288,841);
// by ID

G=gap.SmallGroup(288,841);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,421,219,100,4037,4716,2371,201,10982,4717,3156,622]);
// 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=a*d^3>;
// generators/relations

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