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## G = C3×SL2(𝔽3)  order 72 = 23·32

### Direct product of C3 and SL2(𝔽3)

Aliases: C3×SL2(𝔽3), Q8⋊C32, C6.3A4, C2.(C3×A4), (C3×Q8)⋊C3, SmallGroup(72,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3)
 Lower central Q8 — C3×SL2(𝔽3)
 Upper central C1 — C6

Generators and relations for C3×SL2(𝔽3)
G = < a,b,c,d | a3=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Character table of C3×SL2(𝔽3)

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 4 6A 6B 6C 6D 6E 6F 6G 6H 12A 12B size 1 1 1 1 4 4 4 4 4 4 6 1 1 4 4 4 4 4 4 6 6 ρ1 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 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 1 1 linear of order 3 ρ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 1 ζ32 1 1 ζ32 ζ3 ζ3 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ4 1 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 1 ζ3 1 1 ζ3 ζ32 ζ32 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 1 ζ3 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ32 ζ3 1 ζ3 ζ32 linear of order 3 ρ8 1 1 ζ32 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ3 ζ32 1 ζ32 ζ3 linear of order 3 ρ9 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 1 1 linear of order 3 ρ10 2 -2 2 2 -1 -1 -1 -1 -1 -1 0 -2 -2 1 1 1 1 1 1 0 0 symplectic lifted from SL2(𝔽3), Schur index 2 ρ11 2 -2 -1+√-3 -1-√-3 ζ65 ζ6 -1 ζ6 -1 ζ65 0 1-√-3 1+√-3 ζ3 ζ32 ζ3 1 1 ζ32 0 0 complex faithful ρ12 2 -2 -1-√-3 -1+√-3 ζ6 ζ65 -1 ζ65 -1 ζ6 0 1+√-3 1-√-3 ζ32 ζ3 ζ32 1 1 ζ3 0 0 complex faithful ρ13 2 -2 -1+√-3 -1-√-3 ζ6 ζ65 ζ6 -1 ζ65 -1 0 1-√-3 1+√-3 ζ32 1 1 ζ3 ζ32 ζ3 0 0 complex faithful ρ14 2 -2 2 2 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 0 -2 -2 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 0 0 complex lifted from SL2(𝔽3) ρ15 2 -2 -1+√-3 -1-√-3 -1 -1 ζ65 ζ65 ζ6 ζ6 0 1-√-3 1+√-3 1 ζ3 ζ32 ζ32 ζ3 1 0 0 complex faithful ρ16 2 -2 2 2 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 0 -2 -2 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 0 0 complex lifted from SL2(𝔽3) ρ17 2 -2 -1-√-3 -1+√-3 -1 -1 ζ6 ζ6 ζ65 ζ65 0 1+√-3 1-√-3 1 ζ32 ζ3 ζ3 ζ32 1 0 0 complex faithful ρ18 2 -2 -1-√-3 -1+√-3 ζ65 ζ6 ζ65 -1 ζ6 -1 0 1+√-3 1-√-3 ζ3 1 1 ζ32 ζ3 ζ32 0 0 complex faithful ρ19 3 3 3 3 0 0 0 0 0 0 -1 3 3 0 0 0 0 0 0 -1 -1 orthogonal lifted from A4 ρ20 3 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 ζ6 ζ65 complex lifted from C3×A4 ρ21 3 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 ζ65 ζ6 complex lifted from C3×A4

Permutation representations of C3×SL2(𝔽3)
On 24 points - transitive group 24T70
Generators in S24
(1 8 18)(2 5 19)(3 6 20)(4 7 17)(9 23 15)(10 24 16)(11 21 13)(12 22 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)
(1 12 3 10)(2 11 4 9)(5 21 7 23)(6 24 8 22)(13 17 15 19)(14 20 16 18)
(1 8 18)(2 21 14)(3 6 20)(4 23 16)(5 13 12)(7 15 10)(9 24 17)(11 22 19)

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

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

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

G:=TransitiveGroup(24,70);

C3×SL2(𝔽3) is a maximal subgroup of   C6.5S4  C6.6S4  Dic3.A4  C18.A4  Q8⋊He3  Ω4+ (𝔽3)
C3×SL2(𝔽3) is a maximal quotient of   C18.A4  Q8⋊3- 1+2  Q8⋊He3

Matrix representation of C3×SL2(𝔽3) in GL2(𝔽7) generated by

 4 0 0 4
,
 3 2 2 4
,
 0 1 6 0
,
 2 0 4 1
G:=sub<GL(2,GF(7))| [4,0,0,4],[3,2,2,4],[0,6,1,0],[2,4,0,1] >;

C3×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_3\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3xSL(2,3)");
// GroupNames label

G:=SmallGroup(72,25);
// by ID

G=gap.SmallGroup(72,25);
# by ID

G:=PCGroup([5,-3,-3,-2,2,-2,272,72,543,133,58]);
// Polycyclic

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