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## G = C4×SL2(𝔽3)  order 96 = 25·3

### Direct product of C4 and SL2(𝔽3)

Aliases: C4×SL2(𝔽3), Q8⋊C12, (C4×Q8)⋊C3, C2.2(C4×A4), (C2×C4).1A4, C2.(C4.A4), (C2×Q8).1C6, C22.6(C2×A4), C2.(C2×SL2(𝔽3)), (C2×SL2(𝔽3)).3C2, SmallGroup(96,69)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C4×SL2(𝔽3)
 Chief series C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C4×SL2(𝔽3)
 Lower central Q8 — C4×SL2(𝔽3)
 Upper central C1 — C2×C4

Generators and relations for C4×SL2(𝔽3)
G = < a,b,c,d | a4=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Character table of C4×SL2(𝔽3)

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 4 4 1 1 1 1 6 6 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 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 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 -1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ4 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 1 1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 linear of order 6 ρ5 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 1 1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 linear of order 6 ρ7 1 -1 -1 1 1 1 i i -i -i -1 1 i -i 1 1 -1 -1 -1 -1 -i i -i -i i i -i i linear of order 4 ρ8 1 -1 -1 1 1 1 -i -i i i -1 1 -i i 1 1 -1 -1 -1 -1 i -i i i -i -i i -i linear of order 4 ρ9 1 -1 -1 1 ζ3 ζ32 -i -i i i -1 1 -i i ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ43ζ3 linear of order 12 ρ10 1 -1 -1 1 ζ3 ζ32 i i -i -i -1 1 i -i ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ4ζ3 linear of order 12 ρ11 1 -1 -1 1 ζ32 ζ3 -i -i i i -1 1 -i i ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ43ζ32 linear of order 12 ρ12 1 -1 -1 1 ζ32 ζ3 i i -i -i -1 1 i -i ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ4ζ32 linear of order 12 ρ13 2 -2 2 -2 -1 -1 2 -2 2 -2 0 0 0 0 1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ14 2 -2 2 -2 -1 -1 -2 2 -2 2 0 0 0 0 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ15 2 -2 2 -2 ζ65 ζ6 -2 2 -2 2 0 0 0 0 ζ32 ζ3 ζ6 ζ65 ζ3 ζ32 ζ6 ζ65 ζ3 ζ65 ζ32 ζ6 ζ32 ζ3 complex lifted from SL2(𝔽3) ρ16 2 -2 2 -2 ζ6 ζ65 -2 2 -2 2 0 0 0 0 ζ3 ζ32 ζ65 ζ6 ζ32 ζ3 ζ65 ζ6 ζ32 ζ6 ζ3 ζ65 ζ3 ζ32 complex lifted from SL2(𝔽3) ρ17 2 -2 2 -2 ζ6 ζ65 2 -2 2 -2 0 0 0 0 ζ3 ζ32 ζ65 ζ6 ζ32 ζ3 ζ3 ζ32 ζ6 ζ32 ζ65 ζ3 ζ65 ζ6 complex lifted from SL2(𝔽3) ρ18 2 -2 2 -2 ζ65 ζ6 2 -2 2 -2 0 0 0 0 ζ32 ζ3 ζ6 ζ65 ζ3 ζ32 ζ32 ζ3 ζ65 ζ3 ζ6 ζ32 ζ6 ζ65 complex lifted from SL2(𝔽3) ρ19 2 2 -2 -2 -1 -1 -2i 2i 2i -2i 0 0 0 0 1 1 1 1 -1 -1 i -i -i i i -i -i i complex lifted from C4.A4 ρ20 2 2 -2 -2 -1 -1 2i -2i -2i 2i 0 0 0 0 1 1 1 1 -1 -1 -i i i -i -i i i -i complex lifted from C4.A4 ρ21 2 2 -2 -2 ζ65 ζ6 2i -2i -2i 2i 0 0 0 0 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 complex lifted from C4.A4 ρ22 2 2 -2 -2 ζ65 ζ6 -2i 2i 2i -2i 0 0 0 0 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 complex lifted from C4.A4 ρ23 2 2 -2 -2 ζ6 ζ65 2i -2i -2i 2i 0 0 0 0 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 complex lifted from C4.A4 ρ24 2 2 -2 -2 ζ6 ζ65 -2i 2i 2i -2i 0 0 0 0 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 complex lifted from C4.A4 ρ25 3 3 3 3 0 0 -3 -3 -3 -3 -1 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ26 3 3 3 3 0 0 3 3 3 3 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ27 3 -3 -3 3 0 0 3i 3i -3i -3i 1 -1 -i i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4×A4 ρ28 3 -3 -3 3 0 0 -3i -3i 3i 3i 1 -1 i -i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4×A4

Smallest permutation representation of C4×SL2(𝔽3)
On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 21 20 28)(2 22 17 25)(3 23 18 26)(4 24 19 27)(5 16 30 9)(6 13 31 10)(7 14 32 11)(8 15 29 12)
(1 15 20 12)(2 16 17 9)(3 13 18 10)(4 14 19 11)(5 25 30 22)(6 26 31 23)(7 27 32 24)(8 28 29 21)
(5 16 22)(6 13 23)(7 14 24)(8 15 21)(9 25 30)(10 26 31)(11 27 32)(12 28 29)

G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,21,20,28)(2,22,17,25)(3,23,18,26)(4,24,19,27)(5,16,30,9)(6,13,31,10)(7,14,32,11)(8,15,29,12), (1,15,20,12)(2,16,17,9)(3,13,18,10)(4,14,19,11)(5,25,30,22)(6,26,31,23)(7,27,32,24)(8,28,29,21), (5,16,22)(6,13,23)(7,14,24)(8,15,21)(9,25,30)(10,26,31)(11,27,32)(12,28,29)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,21,20,28)(2,22,17,25)(3,23,18,26)(4,24,19,27)(5,16,30,9)(6,13,31,10)(7,14,32,11)(8,15,29,12), (1,15,20,12)(2,16,17,9)(3,13,18,10)(4,14,19,11)(5,25,30,22)(6,26,31,23)(7,27,32,24)(8,28,29,21), (5,16,22)(6,13,23)(7,14,24)(8,15,21)(9,25,30)(10,26,31)(11,27,32)(12,28,29) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,21,20,28),(2,22,17,25),(3,23,18,26),(4,24,19,27),(5,16,30,9),(6,13,31,10),(7,14,32,11),(8,15,29,12)], [(1,15,20,12),(2,16,17,9),(3,13,18,10),(4,14,19,11),(5,25,30,22),(6,26,31,23),(7,27,32,24),(8,28,29,21)], [(5,16,22),(6,13,23),(7,14,24),(8,15,21),(9,25,30),(10,26,31),(11,27,32),(12,28,29)])

C4×SL2(𝔽3) is a maximal subgroup of
C2.U2(𝔽3)  Q8⋊Dic6  CSU2(𝔽3)⋊C4  Q8.Dic6  Q8.D12  SL2(𝔽3)⋊Q8  Q8⋊D12  GL2(𝔽3)⋊C4  Q8.2D12  (C2×Q8)⋊C12  C4○D4⋊C12  SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  SL2(𝔽3)⋊3Q8

Matrix representation of C4×SL2(𝔽3) in GL3(𝔽13) generated by

 5 0 0 0 8 0 0 0 8
,
 1 0 0 0 10 4 0 4 3
,
 1 0 0 0 0 12 0 1 0
,
 3 0 0 0 1 0 0 4 3
G:=sub<GL(3,GF(13))| [5,0,0,0,8,0,0,0,8],[1,0,0,0,10,4,0,4,3],[1,0,0,0,0,1,0,12,0],[3,0,0,0,1,4,0,0,3] >;

C4×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_4\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C4xSL(2,3)");
// GroupNames label

G:=SmallGroup(96,69);
// by ID

G=gap.SmallGroup(96,69);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-2,36,297,117,550,202,88]);
// Polycyclic

G:=Group<a,b,c,d|a^4=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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