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G = C2.AΓL1(𝔽9)  order 288 = 25·32

1st central extension by C2 of AΓL1(𝔽9)

non-abelian, soluble, monomial

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 12A2B2C2D2E34A4B4C4D6A6B6C8A8B8C8D
 size 119912128181836368242418181818
ρ1111111111111111111    trivial
ρ2111111111-1-1111-1-1-1-1    linear of order 2
ρ31111-1-1111-1-11-1-11111    linear of order 2
ρ41111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ51-1-111-11-11-ii-1-11-i-iii    linear of order 4
ρ61-1-111-11-11i-i-1-11ii-i-i    linear of order 4
ρ71-1-11-111-11i-i-11-1-i-iii    linear of order 4
ρ81-1-11-111-11-ii-11-1ii-i-i    linear of order 4
ρ92-2-220022-200-2000000    orthogonal lifted from D4
ρ102222002-2-2002000000    orthogonal lifted from D4
ρ112-22-20020000-200√2-√2-√2√2    orthogonal lifted from D8
ρ122-22-20020000-200-√2√2√2-√2    orthogonal lifted from D8
ρ1322-2-20020000200√-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 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)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
01000000
-1-1000000
00-1-10000
00100000
0000-1-100
00001000
00000010
00000001
,
-1-1000000
10000000
00100000
00010000
0000-1-100
00001000
000000-1-1
00000010
,
00001000
00000100
00000010
00000001
00100000
00010000
-10000000
11000000
,
10000000
01000000
00-100000
00110000
000000-10
0000000-1
0000-1000
00000-100

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
 

Export

Character table of C2.AΓL1(𝔽9) in TeX

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