Copied to
clipboard

## 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 [×4], C3, C4 [×3], C22 [×5], S3 [×4], C6 [×3], C8, C2×C4 [×2], D4 [×3], C23, C32, D6 [×6], C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C22×S3, D4⋊C4, C32⋊C4 [×2], C32⋊C4, S32 [×3], S3×C6, C2×C3⋊S3, F9, S3≀C2 [×2], S3≀C2, C2×C32⋊C4, C2×C32⋊C4, C2×S32, C2.PSU3(𝔽2), C2×F9, C2×S3≀C2, C2.AΓL1(𝔽9)
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], 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 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(1 18 22)(2 19 23)(3 24 20)(5 12 16)(6 9 13)(8 11 15)
(2 19 23)(3 20 24)(4 17 21)(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 7)(4 5)(10 19)(11 15)(12 17)(14 23)(16 21)(18 22)

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

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

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

G:=TransitiveGroup(24,680);

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

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

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

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

Export

׿
×
𝔽