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## G = C2×ASL2(𝔽3)  order 432 = 24·33

### Direct product of C2 and ASL2(𝔽3)

Aliases: C2×ASL2(𝔽3), PSU3(𝔽2)⋊C6, C3⋊S3⋊SL2(𝔽3), (C3×C6)⋊SL2(𝔽3), (C2×PSU3(𝔽2))⋊C3, C32⋊(C2×SL2(𝔽3)), C3⋊S3.(C2×A4), (C2×C3⋊S3).A4, SmallGroup(432,735)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — PSU3(𝔽2) — C2×ASL2(𝔽3)
 Chief series C1 — C32 — C3⋊S3 — PSU3(𝔽2) — ASL2(𝔽3) — C2×ASL2(𝔽3)
 Lower central PSU3(𝔽2) — C2×ASL2(𝔽3)
 Upper central C1 — C2

Generators and relations for C2×ASL2(𝔽3)
G = < a,b,c,d,e,f | a2=b3=c3=d4=f3=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, cf=fc, ede-1=d-1, fdf-1=e, fef-1=de >

Character table of C2×ASL2(𝔽3)

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I size 1 1 9 9 8 12 12 24 24 54 54 8 12 12 24 24 36 36 36 36 ρ1 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 linear of order 2 ρ3 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ4 1 -1 -1 1 1 ζ3 ζ32 ζ3 ζ32 1 -1 -1 ζ6 ζ65 ζ6 ζ65 ζ32 ζ3 ζ6 ζ65 linear of order 6 ρ5 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 -1 -1 1 1 ζ32 ζ3 ζ32 ζ3 1 -1 -1 ζ65 ζ6 ζ65 ζ6 ζ3 ζ32 ζ65 ζ6 linear of order 6 ρ7 2 -2 2 -2 2 -1 -1 -1 -1 0 0 -2 1 1 1 1 1 1 -1 -1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ8 2 2 -2 -2 2 -1 -1 -1 -1 0 0 2 -1 -1 -1 -1 1 1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ9 2 -2 2 -2 2 ζ6 ζ65 ζ6 ζ65 0 0 -2 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 complex lifted from SL2(𝔽3) ρ10 2 2 -2 -2 2 ζ65 ζ6 ζ65 ζ6 0 0 2 ζ6 ζ65 ζ6 ζ65 ζ32 ζ3 ζ32 ζ3 complex lifted from SL2(𝔽3) ρ11 2 2 -2 -2 2 ζ6 ζ65 ζ6 ζ65 0 0 2 ζ65 ζ6 ζ65 ζ6 ζ3 ζ32 ζ3 ζ32 complex lifted from SL2(𝔽3) ρ12 2 -2 2 -2 2 ζ65 ζ6 ζ65 ζ6 0 0 -2 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 complex lifted from SL2(𝔽3) ρ13 3 -3 -3 3 3 0 0 0 0 -1 1 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 3 3 3 0 0 0 0 -1 -1 3 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 8 8 0 0 -1 2 2 -1 -1 0 0 -1 2 2 -1 -1 0 0 0 0 orthogonal lifted from ASL2(𝔽3) ρ16 8 -8 0 0 -1 2 2 -1 -1 0 0 1 -2 -2 1 1 0 0 0 0 orthogonal faithful ρ17 8 -8 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 1 1-√-3 1+√-3 ζ3 ζ32 0 0 0 0 complex faithful ρ18 8 8 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 0 0 complex lifted from ASL2(𝔽3) ρ19 8 -8 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 1 1+√-3 1-√-3 ζ32 ζ3 0 0 0 0 complex faithful ρ20 8 8 0 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 0 0 complex lifted from ASL2(𝔽3)

Permutation representations of C2×ASL2(𝔽3)
On 18 points - transitive group 18T151
Generators in S18
(1 2)(3 14)(4 11)(5 12)(6 13)(7 17)(8 18)(9 15)(10 16)
(1 15 17)(2 9 7)(3 8 4)(5 6 10)(11 14 18)(12 13 16)
(1 16 18)(2 10 8)(3 7 6)(4 9 5)(11 15 12)(13 14 17)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(3 8 5 10)(4 7 6 9)(11 17 13 15)(12 16 14 18)
(3 7 6)(4 5 9)(11 12 15)(13 14 17)

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

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

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

G:=TransitiveGroup(18,151);

On 24 points - transitive group 24T1320
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 13)(10 14)(11 15)(12 16)(17 23)(18 24)(19 21)(20 22)
(1 22 24)(2 23 21)(4 13 15)(5 9 11)(7 20 18)(8 17 19)
(1 24 22)(2 23 21)(3 14 16)(6 10 12)(7 18 20)(8 17 19)
(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 6)(3 8)(4 7)(9 24 11 22)(10 23 12 21)(13 18 15 20)(14 17 16 19)
(1 6 2)(3 8 7)(10 23 24)(12 21 22)(14 17 18)(16 19 20)

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

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

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

G:=TransitiveGroup(24,1320);

On 24 points - transitive group 24T1321
Generators in S24
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(8 9 24)
(2 20 15)(4 13 18)(5 10 21)(6 11 22)(7 23 12)(8 24 9)
(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 22 3 24)(2 21 4 23)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)
(2 21 22)(4 23 24)(5 6 20)(7 8 18)(9 13 12)(10 11 15)

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

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

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

G:=TransitiveGroup(24,1321);

Matrix representation of C2×ASL2(𝔽3) 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 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 -1 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 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 -1 -1 -1 -1 -1 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 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1
,
 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 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 0 1 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 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,-1,0,0,0,1,0,0,0,-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],[1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1],[1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C2×ASL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times {\rm ASL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xASL(2,3)");
// GroupNames label

G:=SmallGroup(432,735);
// by ID

G=gap.SmallGroup(432,735);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,-3,3,387,100,262,185,80,6060,1699,1034,201,8245,1588,223,622]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^4=f^3=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,c*f=f*c,e*d*e^-1=d^-1,f*d*f^-1=e,f*e*f^-1=d*e>;
// generators/relations

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