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G = C3×SL2(𝔽3)  order 72 = 23·32

Direct product of C3 and SL2(𝔽3)

direct product, non-abelian, soluble

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

Series: Derived Chief Lower central Upper central

C1C2Q8 — C3×SL2(𝔽3)
C1C2Q8SL2(𝔽3) — C3×SL2(𝔽3)
Q8 — C3×SL2(𝔽3)
C1C6

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 >

4C3
4C3
4C3
3C4
4C6
4C6
4C6
4C32
3C12
4C3×C6

Character table of C3×SL2(𝔽3)

 class 123A3B3C3D3E3F3G3H46A6B6C6D6E6F6G6H12A12B
 size 111144444461144444466
ρ1111111111111111111111    trivial
ρ21111ζ3ζ32ζ32ζ3ζ3ζ32111ζ3ζ3ζ32ζ3ζ32ζ3211    linear of order 3
ρ311ζ32ζ3ζ3ζ32ζ31ζ3211ζ32ζ3ζ311ζ32ζ3ζ32ζ32ζ3    linear of order 3
ρ411ζ3ζ32ζ3ζ321ζ321ζ31ζ3ζ32ζ3ζ32ζ311ζ32ζ3ζ32    linear of order 3
ρ511ζ3ζ32ζ32ζ3ζ321ζ311ζ3ζ32ζ3211ζ3ζ32ζ3ζ3ζ32    linear of order 3
ρ611ζ32ζ3ζ32ζ31ζ31ζ321ζ32ζ3ζ32ζ3ζ3211ζ3ζ32ζ3    linear of order 3
ρ711ζ3ζ3211ζ3ζ3ζ32ζ321ζ3ζ321ζ3ζ32ζ32ζ31ζ3ζ32    linear of order 3
ρ811ζ32ζ311ζ32ζ32ζ3ζ31ζ32ζ31ζ32ζ3ζ3ζ321ζ32ζ3    linear of order 3
ρ91111ζ32ζ3ζ3ζ32ζ32ζ3111ζ32ζ32ζ3ζ32ζ3ζ311    linear of order 3
ρ102-222-1-1-1-1-1-10-2-211111100    symplectic lifted from SL2(𝔽3), Schur index 2
ρ112-2-1+-3-1--3ζ65ζ6-1ζ6-1ζ6501--31+-3ζ3ζ32ζ311ζ3200    complex faithful
ρ122-2-1--3-1+-3ζ6ζ65-1ζ65-1ζ601+-31--3ζ32ζ3ζ3211ζ300    complex faithful
ρ132-2-1+-3-1--3ζ6ζ65ζ6-1ζ65-101--31+-3ζ3211ζ3ζ32ζ300    complex faithful
ρ142-222ζ65ζ6ζ6ζ65ζ65ζ60-2-2ζ3ζ3ζ32ζ3ζ32ζ3200    complex lifted from SL2(𝔽3)
ρ152-2-1+-3-1--3-1-1ζ65ζ65ζ6ζ601--31+-31ζ3ζ32ζ32ζ3100    complex faithful
ρ162-222ζ6ζ65ζ65ζ6ζ6ζ650-2-2ζ32ζ32ζ3ζ32ζ3ζ300    complex lifted from SL2(𝔽3)
ρ172-2-1--3-1+-3-1-1ζ6ζ6ζ65ζ6501+-31--31ζ32ζ3ζ3ζ32100    complex faithful
ρ182-2-1--3-1+-3ζ65ζ6ζ65-1ζ6-101+-31--3ζ311ζ32ζ3ζ3200    complex faithful
ρ193333000000-133000000-1-1    orthogonal lifted from A4
ρ2033-3-3-3/2-3+3-3/2000000-1-3-3-3/2-3+3-3/2000000ζ6ζ65    complex lifted from C3×A4
ρ2133-3+3-3/2-3-3-3/2000000-1-3+3-3/2-3-3-3/2000000ζ65ζ6    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

40
04
,
32
24
,
01
60
,
20
41
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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Subgroup lattice of C3×SL2(𝔽3) in TeX
Character table of C3×SL2(𝔽3) in TeX

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