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G = C2×SL2(𝔽3)  order 48 = 24·3

Direct product of C2 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C2×SL2(𝔽3), Q8⋊C6, C22.2A4, (C2×Q8)⋊C3, C2.2(C2×A4), SmallGroup(48,32)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C2×SL2(𝔽3)
C1C2Q8SL2(𝔽3) — C2×SL2(𝔽3)
Q8 — C2×SL2(𝔽3)
C1C22

Generators and relations for C2×SL2(𝔽3)
 G = < a,b,c,d | a2=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
3C4
3C4
4C6
4C6
4C6
3C2×C4
3Q8
4C2×C6

Character table of C2×SL2(𝔽3)

 class 12A2B2C3A3B4A4B6A6B6C6D6E6F
 size 11114466444444
ρ111111111111111    trivial
ρ21-11-1111-111-1-1-1-1    linear of order 2
ρ31111ζ32ζ311ζ3ζ32ζ3ζ3ζ32ζ32    linear of order 3
ρ41111ζ3ζ3211ζ32ζ3ζ32ζ32ζ3ζ3    linear of order 3
ρ51-11-1ζ32ζ31-1ζ3ζ32ζ65ζ65ζ6ζ6    linear of order 6
ρ61-11-1ζ3ζ321-1ζ32ζ3ζ6ζ6ζ65ζ65    linear of order 6
ρ722-2-2-1-10011-11-11    symplectic lifted from SL2(𝔽3), Schur index 2
ρ82-2-22-1-100111-11-1    symplectic lifted from SL2(𝔽3), Schur index 2
ρ92-2-22ζ6ζ6500ζ3ζ32ζ3ζ65ζ32ζ6    complex lifted from SL2(𝔽3)
ρ1022-2-2ζ65ζ600ζ32ζ3ζ6ζ32ζ65ζ3    complex lifted from SL2(𝔽3)
ρ1122-2-2ζ6ζ6500ζ3ζ32ζ65ζ3ζ6ζ32    complex lifted from SL2(𝔽3)
ρ122-2-22ζ65ζ600ζ32ζ3ζ32ζ6ζ3ζ65    complex lifted from SL2(𝔽3)
ρ133-33-300-11000000    orthogonal lifted from C2×A4
ρ14333300-1-1000000    orthogonal lifted from A4

Permutation representations of C2×SL2(𝔽3)
On 16 points - transitive group 16T59
Generators in S16
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 3 10)(2 11 4 9)(5 13 7 15)(6 16 8 14)
(2 11 12)(4 9 10)(5 13 14)(7 15 16)

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

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

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

G:=TransitiveGroup(16,59);

Matrix representation of C2×SL2(𝔽3) in GL3(𝔽13) generated by

1200
0120
0012
,
100
015
01012
,
100
0122
0121
,
300
010
0103
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[1,0,0,0,1,10,0,5,12],[1,0,0,0,12,12,0,2,1],[3,0,0,0,1,10,0,0,3] >;

C2×SL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C2xSL(2,3)");
// GroupNames label

G:=SmallGroup(48,32);
// by ID

G=gap.SmallGroup(48,32);
# by ID

G:=PCGroup([5,-2,-3,-2,2,-2,97,72,188,133,58]);
// Polycyclic

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

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