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## G = C3×AΓL1(𝔽9)  order 432 = 24·33

### Direct product of C3 and AΓL1(𝔽9)

Aliases: C3×AΓL1(𝔽9), F9⋊C6, C331SD16, PSU3(𝔽2)⋊2C6, S3≀C2.C6, (C3×F9)⋊3C2, C32⋊(C3×SD16), (C3×PSU3(𝔽2))⋊3C2, C3⋊S3.(C3×D4), C32⋊C4.(C2×C6), (C3×C3⋊S3).1D4, (C3×S3≀C2).2C2, (C3×C32⋊C4).6C22, SmallGroup(432,737)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C4 — C3×AΓL1(𝔽9)
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C3×C32⋊C4 — C3×F9 — C3×AΓL1(𝔽9)
 Lower central C32 — C3⋊S3 — C32⋊C4 — C3×AΓL1(𝔽9)
 Upper central C1 — C3

Generators and relations for C3×AΓL1(𝔽9)
G = < a,b,c,d,e | a3=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 >

Character table of C3×AΓL1(𝔽9)

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 12A 12B 12C 12D 24A 24B 24C 24D size 1 9 12 1 1 8 8 8 18 36 9 9 12 12 24 24 24 18 18 18 18 36 36 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 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 -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 -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 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 1 ζ3 ζ32 1 ζ32 ζ3 1 -1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ6 1 1 1 ζ3 ζ32 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 1 1 ζ32 ζ3 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ8 1 1 -1 ζ32 ζ3 1 ζ3 ζ32 1 -1 ζ3 ζ32 ζ65 ζ6 -1 ζ6 ζ65 1 1 ζ32 ζ3 ζ6 ζ65 ζ32 ζ3 ζ3 ζ32 linear of order 6 ρ9 1 1 -1 ζ3 ζ32 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ6 ζ65 -1 ζ65 ζ6 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ10 1 1 -1 ζ32 ζ3 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ65 ζ6 -1 ζ6 ζ65 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ11 1 1 1 ζ32 ζ3 1 ζ3 ζ32 1 -1 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ12 1 1 -1 ζ3 ζ32 1 ζ32 ζ3 1 -1 ζ32 ζ3 ζ6 ζ65 -1 ζ65 ζ6 1 1 ζ3 ζ32 ζ65 ζ6 ζ3 ζ32 ζ32 ζ3 linear of order 6 ρ13 2 2 0 2 2 2 2 2 -2 0 2 2 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 0 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 -2 0 -1+√-3 -1-√-3 0 0 0 0 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 complex lifted from C3×D4 ρ15 2 2 0 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 -2 0 -1-√-3 -1+√-3 0 0 0 0 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 complex lifted from C3×D4 ρ16 2 -2 0 2 2 2 2 2 0 0 -2 -2 0 0 0 0 0 -√-2 √-2 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 0 2 2 2 2 2 0 0 -2 -2 0 0 0 0 0 √-2 -√-2 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 0 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 0 0 1+√-3 1-√-3 0 0 0 0 0 √-2 -√-2 0 0 0 0 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 complex lifted from C3×SD16 ρ19 2 -2 0 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 0 0 1-√-3 1+√-3 0 0 0 0 0 -√-2 √-2 0 0 0 0 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 complex lifted from C3×SD16 ρ20 2 -2 0 -1+√-3 -1-√-3 2 -1-√-3 -1+√-3 0 0 1+√-3 1-√-3 0 0 0 0 0 -√-2 √-2 0 0 0 0 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 complex lifted from C3×SD16 ρ21 2 -2 0 -1-√-3 -1+√-3 2 -1+√-3 -1-√-3 0 0 1-√-3 1+√-3 0 0 0 0 0 √-2 -√-2 0 0 0 0 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 complex lifted from C3×SD16 ρ22 8 0 -2 8 8 -1 -1 -1 0 0 0 0 -2 -2 1 1 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ23 8 0 2 8 8 -1 -1 -1 0 0 0 0 2 2 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from AΓL1(𝔽9) ρ24 8 0 -2 -4+4√-3 -4-4√-3 -1 ζ6 ζ65 0 0 0 0 1+√-3 1-√-3 1 ζ3 ζ32 0 0 0 0 0 0 0 0 0 0 complex faithful ρ25 8 0 -2 -4-4√-3 -4+4√-3 -1 ζ65 ζ6 0 0 0 0 1-√-3 1+√-3 1 ζ32 ζ3 0 0 0 0 0 0 0 0 0 0 complex faithful ρ26 8 0 2 -4+4√-3 -4-4√-3 -1 ζ6 ζ65 0 0 0 0 -1-√-3 -1+√-3 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 complex faithful ρ27 8 0 2 -4-4√-3 -4+4√-3 -1 ζ65 ζ6 0 0 0 0 -1+√-3 -1-√-3 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×AΓL1(𝔽9)
On 24 points - transitive group 24T1330
Generators in S24
(1 9 21)(2 10 22)(3 11 23)(4 12 24)(5 13 17)(6 14 18)(7 15 19)(8 16 20)
(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 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,1330);

On 27 points - transitive group 27T142
Generators in S27
(1 3 2)(4 19 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 20)(10 17 21)(11 18 22)
(1 17 13)(2 10 6)(3 21 25)(4 5 7)(8 11 9)(12 14 19)(15 18 16)(20 27 22)(23 24 26)
(1 18 14)(2 11 7)(3 22 26)(4 10 9)(5 6 8)(12 13 15)(16 19 17)(20 23 21)(24 25 27)
(4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27)
(4 6)(5 9)(8 10)(12 16)(13 19)(15 17)(20 24)(21 27)(23 25)

G:=sub<Sym(27)| (1,3,2)(4,19,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,20)(10,17,21)(11,18,22), (1,17,13)(2,10,6)(3,21,25)(4,5,7)(8,11,9)(12,14,19)(15,18,16)(20,27,22)(23,24,26), (1,18,14)(2,11,7)(3,22,26)(4,10,9)(5,6,8)(12,13,15)(16,19,17)(20,23,21)(24,25,27), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27), (4,6)(5,9)(8,10)(12,16)(13,19)(15,17)(20,24)(21,27)(23,25)>;

G:=Group( (1,3,2)(4,19,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,20)(10,17,21)(11,18,22), (1,17,13)(2,10,6)(3,21,25)(4,5,7)(8,11,9)(12,14,19)(15,18,16)(20,27,22)(23,24,26), (1,18,14)(2,11,7)(3,22,26)(4,10,9)(5,6,8)(12,13,15)(16,19,17)(20,23,21)(24,25,27), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27), (4,6)(5,9)(8,10)(12,16)(13,19)(15,17)(20,24)(21,27)(23,25) );

G=PermutationGroup([(1,3,2),(4,19,23),(5,12,24),(6,13,25),(7,14,26),(8,15,27),(9,16,20),(10,17,21),(11,18,22)], [(1,17,13),(2,10,6),(3,21,25),(4,5,7),(8,11,9),(12,14,19),(15,18,16),(20,27,22),(23,24,26)], [(1,18,14),(2,11,7),(3,22,26),(4,10,9),(5,6,8),(12,13,15),(16,19,17),(20,23,21),(24,25,27)], [(4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27)], [(4,6),(5,9),(8,10),(12,16),(13,19),(15,17),(20,24),(21,27),(23,25)])

G:=TransitiveGroup(27,142);

Matrix representation of C3×AΓL1(𝔽9) in GL8(𝔽73)

 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 64
,
 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 0 0 64 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 8 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 64
,
 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 0 1 0 0 0 0 0 0 1 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 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 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C3×AΓL1(𝔽9) in GAP, Magma, Sage, TeX

C_3\times {\rm AGammaL}_1({\mathbb F}_9)
% in TeX

G:=Group("C3xAGammaL(1,9)");
// GroupNames label

G:=SmallGroup(432,737);
// by ID

G=gap.SmallGroup(432,737);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,533,514,80,6053,7068,1202,201,16470,7069,1595,622]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=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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