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## G = C23.SL2(𝔽3)  order 192 = 26·3

### 1st non-split extension by C23 of SL2(𝔽3) acting via SL2(𝔽3)/C2=A4

Aliases: C23.1SL2(𝔽3), C4.10C42⋊C3, C4.2(C42⋊C3), (C22×C4).6A4, C2.(C23.3A4), SmallGroup(192,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C4.10C42 — C23.SL2(𝔽3)
 Chief series C1 — C2 — C4 — C22×C4 — C4.10C42 — C23.SL2(𝔽3)
 Lower central C4.10C42 — C23.SL2(𝔽3)
 Upper central C1 — C4

Generators and relations for C23.SL2(𝔽3)
G = < a,b,c,d,e,f | a2=b2=c2=f3=1, d4=c, e2=bd2, ab=ba, eae-1=fbf-1=ac=ca, ad=da, faf-1=abc, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, cf=fc, ede-1=ad3, fdf-1=be, fef-1=acde >

Character table of C23.SL2(𝔽3)

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D size 1 1 6 16 16 1 1 6 16 16 12 12 12 12 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ3 1 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 2 2 2 -1 -1 -2 -2 -2 -1 -1 0 0 0 0 1 1 1 1 symplectic lifted from SL2(𝔽3), Schur index 2 ρ5 2 2 2 ζ6 ζ65 -2 -2 -2 ζ65 ζ6 0 0 0 0 ζ3 ζ3 ζ32 ζ32 complex lifted from SL2(𝔽3) ρ6 2 2 2 ζ65 ζ6 -2 -2 -2 ζ6 ζ65 0 0 0 0 ζ32 ζ32 ζ3 ζ3 complex lifted from SL2(𝔽3) ρ7 3 3 3 0 0 3 3 3 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from A4 ρ8 3 3 -1 0 0 3 3 -1 0 0 -1-2i 1 -1+2i 1 0 0 0 0 complex lifted from C42⋊C3 ρ9 3 3 -1 0 0 3 3 -1 0 0 1 -1-2i 1 -1+2i 0 0 0 0 complex lifted from C42⋊C3 ρ10 3 3 -1 0 0 3 3 -1 0 0 1 -1+2i 1 -1-2i 0 0 0 0 complex lifted from C42⋊C3 ρ11 3 3 -1 0 0 3 3 -1 0 0 -1+2i 1 -1-2i 1 0 0 0 0 complex lifted from C42⋊C3 ρ12 4 -4 0 1 1 4i -4i 0 -1 -1 0 0 0 0 i -i i -i complex faithful ρ13 4 -4 0 1 1 -4i 4i 0 -1 -1 0 0 0 0 -i i -i i complex faithful ρ14 4 -4 0 ζ3 ζ32 -4i 4i 0 ζ6 ζ65 0 0 0 0 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 complex faithful ρ15 4 -4 0 ζ32 ζ3 -4i 4i 0 ζ65 ζ6 0 0 0 0 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 complex faithful ρ16 4 -4 0 ζ3 ζ32 4i -4i 0 ζ6 ζ65 0 0 0 0 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 complex faithful ρ17 4 -4 0 ζ32 ζ3 4i -4i 0 ζ65 ζ6 0 0 0 0 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 complex faithful ρ18 6 6 -2 0 0 -6 -6 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23.3A4

Permutation representations of C23.SL2(𝔽3)
On 16 points - transitive group 16T439
Generators in S16
(9 13)(10 14)(11 15)(12 16)
(1 5)(3 7)(10 14)(12 16)
(1 5)(2 6)(3 7)(4 8)(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 11 7 13 5 15 3 9)(2 14 4 12 6 10 8 16)
(2 14 15)(4 12 13)(6 10 11)(8 16 9)

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

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

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

G:=TransitiveGroup(16,439);

Matrix representation of C23.SL2(𝔽3) in GL4(𝔽5) generated by

 1 0 0 0 0 1 0 0 0 0 4 0 0 0 0 4
,
 4 0 0 0 0 1 0 0 0 0 1 0 0 0 0 4
,
 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 4 0 0 2 0 0 0 0 0 0 3 0 0 4 0
,
 0 0 2 0 0 0 0 2 1 0 0 0 0 4 0 0
,
 1 0 0 0 0 0 3 0 0 0 0 3 0 4 0 0
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,4,0,0,0,0,0,0,4,0,0,3,0],[0,0,1,0,0,0,0,4,2,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,4,0,3,0,0,0,0,3,0] >;

C23.SL2(𝔽3) in GAP, Magma, Sage, TeX

C_2^3.{\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2^3.SL(2,3)");
// GroupNames label

G:=SmallGroup(192,4);
// by ID

G=gap.SmallGroup(192,4);
# by ID

G:=PCGroup([7,-3,-2,2,-2,2,-2,-2,85,176,695,387,58,4707,1018,248,2944,1411,718,124]);
// Polycyclic

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

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