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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)

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)

Generators and relations
 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 >

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

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

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

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

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

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

(1 18 22)(3 20 24)(4 21 17)(5 14 10)(6 11 15)(8 13 9)

(1 22 18)(2 19 23)(4 21 17)(5 14 10)(6 15 11)(7 12 16)

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

(1 12 21)(2 13 22)(3 23 14)(5 17 16)(6 18 9)(7 10 19)

(2 13 22)(3 14 23)(4 24 15)(6 18 9)(7 19 10)(8 11 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 15)(11 13)(12 16)(17 21)(18 24)(20 22)
 
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), (1,12,21)(2,13,22)(3,23,14)(5,17,16)(6,18,9)(7,10,19), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (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)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22)>;
 
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), (1,12,21)(2,13,22)(3,23,14)(5,17,16)(6,18,9)(7,10,19), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (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)(9,15)(11,13)(12,16)(17,21)(18,24)(20,22) );
 
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)], [(1,12,21),(2,13,22),(3,23,14),(5,17,16),(6,18,9),(7,10,19)], [(2,13,22),(3,14,23),(4,24,15),(6,18,9),(7,19,10),(8,11,20)], [(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),(9,15),(11,13),(12,16),(17,21),(18,24),(20,22)])
 
G:=TransitiveGroup(24,682);
 

Matrix representation ►G ⊆ 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] >;
 

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

In GAP, Magma, Sage, TeX 

C_2\times 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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