Copied to
clipboard

G = C23.SL2(𝔽3)  order 192 = 26·3

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

non-abelian, soluble

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

C1C4C4.10C42 — C23.SL2(𝔽3)
C1C2C4C22×C4C4.10C42 — C23.SL2(𝔽3)
C4.10C42 — C23.SL2(𝔽3)
C1C4

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 >

6C2
16C3
3C4
3C22
4C22
16C6
3C2×C4
3C2×C4
6C8
6C8
4A4
16C12
3C2×C8
3C2×C8
6M4(2)
6M4(2)
4C2×A4
3C2×M4(2)
4C4×A4

Character table of C23.SL2(𝔽3)

 class 12A2B3A3B4A4B4C6A6B8A8B8C8D12A12B12C12D
 size 116161611616161212121216161616
ρ1111111111111111111    trivial
ρ2111ζ3ζ32111ζ32ζ31111ζ32ζ32ζ3ζ3    linear of order 3
ρ3111ζ32ζ3111ζ3ζ321111ζ3ζ3ζ32ζ32    linear of order 3
ρ4222-1-1-2-2-2-1-100001111    symplectic lifted from SL2(𝔽3), Schur index 2
ρ5222ζ6ζ65-2-2-2ζ65ζ60000ζ3ζ3ζ32ζ32    complex lifted from SL2(𝔽3)
ρ6222ζ65ζ6-2-2-2ζ6ζ650000ζ32ζ32ζ3ζ3    complex lifted from SL2(𝔽3)
ρ73330033300-1-1-1-10000    orthogonal lifted from A4
ρ833-10033-100-1-2i1-1+2i10000    complex lifted from C42⋊C3
ρ933-10033-1001-1-2i1-1+2i0000    complex lifted from C42⋊C3
ρ1033-10033-1001-1+2i1-1-2i0000    complex lifted from C42⋊C3
ρ1133-10033-100-1+2i1-1-2i10000    complex lifted from C42⋊C3
ρ124-40114i-4i0-1-10000i-ii-i    complex faithful
ρ134-4011-4i4i0-1-10000-ii-ii    complex faithful
ρ144-40ζ3ζ32-4i4i0ζ6ζ650000ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex faithful
ρ154-40ζ32ζ3-4i4i0ζ65ζ60000ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex faithful
ρ164-40ζ3ζ324i-4i0ζ6ζ650000ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex faithful
ρ174-40ζ32ζ34i-4i0ζ65ζ60000ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex faithful
ρ1866-200-6-620000000000    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

1000
0100
0040
0004
,
4000
0100
0010
0004
,
4000
0400
0040
0004
,
0400
2000
0003
0040
,
0020
0002
1000
0400
,
1000
0030
0003
0400
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

Export

Subgroup lattice of C23.SL2(𝔽3) in TeX
Character table of C23.SL2(𝔽3) in TeX

׿
×
𝔽