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G = C2×AΓL1(𝔽9)  order 288 = 25·32

Direct product of C2 and AΓL1(𝔽9)

direct product, non-abelian, soluble, monomial

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, C3, C4, C22, S3, C6, C8, C2×C4, D4, Q8, C23, C32, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3⋊S3, C3×C6, C22×S3, C2×SD16, C32⋊C4, C32⋊C4, S32, S3×C6, C2×C3⋊S3, F9, S3≀C2, S3≀C2, PSU3(𝔽2), PSU3(𝔽2), C2×C32⋊C4, C2×C32⋊C4, C2×S32, AΓL1(𝔽9), C2×F9, C2×S3≀C2, C2×PSU3(𝔽2), C2×AΓL1(𝔽9)
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C2×SD16, AΓL1(𝔽9), C2×AΓL1(𝔽9)

Character table of C2×AΓL1(𝔽9)

 class 12A2B2C2D2E34A4B4C4D6A6B6C8A8B8C8D
 size 119912128181836368242418181818
ρ1111111111111111111    trivial
ρ21-11-1-1111-11-1-11-1-1-111    linear of order 2
ρ3111111111-1-1111-1-1-1-1    linear of order 2
ρ41111-1-1111-1-11-1-11111    linear of order 2
ρ51-11-11-111-11-1-1-1111-1-1    linear of order 2
ρ61-11-1-1111-1-11-11-111-1-1    linear of order 2
ρ71-11-11-111-1-11-1-11-1-111    linear of order 2
ρ81111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ92222002-2-2002000000    orthogonal lifted from D4
ρ102-22-2002-2200-2000000    orthogonal lifted from D4
ρ112-2-220020000-200√-2-√-2-√-2√-2    complex lifted from SD16
ρ1222-2-20020000200√-2-√-2√-2-√-2    complex lifted from SD16
ρ132-2-220020000-200-√-2√-2√-2-√-2    complex lifted from SD16
ρ1422-2-20020000200-√-2√-2-√-2√-2    complex lifted from SD16
ρ158-8002-2-1000011-10000    orthogonal faithful
ρ168800-2-2-10000-1110000    orthogonal lifted from AΓL1(𝔽9)
ρ17880022-10000-1-1-10000    orthogonal lifted from AΓL1(𝔽9)
ρ188-800-22-100001-110000    orthogonal faithful

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

(1 15 11)(2 8 4)(3 5 10)(6 9 7)(12 17 18)(13 16 14)

(1 16 12)(2 9 5)(3 4 6)(7 10 8)(11 13 18)(14 17 15)

(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)

(3 7)(4 10)(6 8)(11 17)(13 15)(14 18)
 
G:=sub<Sym(18)| (1,2)(3,18)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17), (1,15,11)(2,8,4)(3,5,10)(6,9,7)(12,17,18)(13,16,14), (1,16,12)(2,9,5)(3,4,6)(7,10,8)(11,13,18)(14,17,15), (3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (3,7)(4,10)(6,8)(11,17)(13,15)(14,18)>;
 
G:=Group( (1,2)(3,18)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17), (1,15,11)(2,8,4)(3,5,10)(6,9,7)(12,17,18)(13,16,14), (1,16,12)(2,9,5)(3,4,6)(7,10,8)(11,13,18)(14,17,15), (3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18), (3,7)(4,10)(6,8)(11,17)(13,15)(14,18) );
 
G=PermutationGroup([[(1,2),(3,18),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(10,17)], [(1,15,11),(2,8,4),(3,5,10),(6,9,7),(12,17,18),(13,16,14)], [(1,16,12),(2,9,5),(3,4,6),(7,10,8),(11,13,18),(14,17,15)], [(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)], [(3,7),(4,10),(6,8),(11,17),(13,15),(14,18)]])
 
G:=TransitiveGroup(18,110);
 

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

(1 19 23)(3 21 17)(4 22 18)(5 12 16)(6 9 13)(8 11 15)

(1 23 19)(2 20 24)(4 22 18)(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)

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

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00000001
-1-1-1-1-1-1-1-1
00000010
01000000
00100000
10000000
00001000
00000100
,
00001000
00000100
00010000
00000001
-1-1-1-1-1-1-1-1
00000010
01000000
00100000
,
-10000000
000-10000
000000-10
0000-1000
0000000-1
0-1000000
11111111
00-100000
,
-10000000
00-100000
0-1000000
00000-100
0000-1000
000-10000
0000000-1
000000-10

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
 

Export

Character table of C2×AΓL1(𝔽9) in TeX

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