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## G = S3×M4(2)  order 96 = 25·3

### Direct product of S3 and M4(2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×M4(2)
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×M4(2)
 Lower central C3 — C6 — S3×M4(2)
 Upper central C1 — C4 — M4(2)

Generators and relations for S3×M4(2)
G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 130 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, D6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, C2×M4(2), S3×C8, C8⋊S3, C4.Dic3, C3×M4(2), S3×C2×C4, S3×M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4×S3, C22×S3, C2×M4(2), S3×C2×C4, S3×M4(2)

Character table of S3×M4(2)

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 24A 24B 24C 24D size 1 1 2 3 3 6 2 1 1 2 3 3 6 2 4 2 2 2 2 6 6 6 6 2 2 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 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 -1 -1 linear of order 2 ρ3 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 1 1 linear of order 2 ρ4 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 -1 -1 linear of order 2 ρ5 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 1 -1 linear of order 2 ρ6 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 -1 1 linear of order 2 ρ7 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 1 -1 linear of order 2 ρ8 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 -1 1 linear of order 2 ρ9 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 i -i i -i -i i i -i -1 -1 1 -i -i i i linear of order 4 ρ10 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 i -i -i i i i -i -i -1 -1 -1 i -i -i i linear of order 4 ρ11 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -i i i -i i i -i -i -1 -1 -1 -i i i -i linear of order 4 ρ12 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 -i i -i i -i i i -i -1 -1 1 i i -i -i linear of order 4 ρ13 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i i i -i -i -i i i -1 -1 -1 -i i i -i linear of order 4 ρ14 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 -i i -i i i -i -i i -1 -1 1 i i -i -i linear of order 4 ρ15 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 i -i i -i i -i -i i -1 -1 1 -i -i i i linear of order 4 ρ16 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 i -i -i i -i -i i i -1 -1 -1 i -i -i i linear of order 4 ρ17 2 2 -2 0 0 0 -1 2 2 -2 0 0 0 -1 1 -2 -2 2 2 0 0 0 0 -1 -1 1 -1 1 -1 1 orthogonal lifted from D6 ρ18 2 2 2 0 0 0 -1 2 2 2 0 0 0 -1 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ19 2 2 -2 0 0 0 -1 2 2 -2 0 0 0 -1 1 2 2 -2 -2 0 0 0 0 -1 -1 1 1 -1 1 -1 orthogonal lifted from D6 ρ20 2 2 2 0 0 0 -1 2 2 2 0 0 0 -1 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ21 2 2 -2 0 0 0 -1 -2 -2 2 0 0 0 -1 1 2i -2i 2i -2i 0 0 0 0 1 1 -1 i i -i -i complex lifted from C4×S3 ρ22 2 -2 0 2 -2 0 2 -2i 2i 0 2i -2i 0 -2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex lifted from M4(2) ρ23 2 -2 0 -2 2 0 2 2i -2i 0 2i -2i 0 -2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex lifted from M4(2) ρ24 2 2 2 0 0 0 -1 -2 -2 -2 0 0 0 -1 -1 2i -2i -2i 2i 0 0 0 0 1 1 1 -i i i -i complex lifted from C4×S3 ρ25 2 2 -2 0 0 0 -1 -2 -2 2 0 0 0 -1 1 -2i 2i -2i 2i 0 0 0 0 1 1 -1 -i -i i i complex lifted from C4×S3 ρ26 2 -2 0 -2 2 0 2 -2i 2i 0 -2i 2i 0 -2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex lifted from M4(2) ρ27 2 2 2 0 0 0 -1 -2 -2 -2 0 0 0 -1 -1 -2i 2i 2i -2i 0 0 0 0 1 1 1 i -i -i i complex lifted from C4×S3 ρ28 2 -2 0 2 -2 0 2 2i -2i 0 -2i 2i 0 -2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex lifted from M4(2) ρ29 4 -4 0 0 0 0 -2 4i -4i 0 0 0 0 2 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 0 complex faithful ρ30 4 -4 0 0 0 0 -2 -4i 4i 0 0 0 0 2 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 0 complex faithful

Permutation representations of S3×M4(2)
On 24 points - transitive group 24T104
Generators in S24
(1 9 22)(2 10 23)(3 11 24)(4 12 17)(5 13 18)(6 14 19)(7 15 20)(8 16 21)
(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)

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

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

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

G:=TransitiveGroup(24,104);

Matrix representation of S3×M4(2) in GL4(𝔽5) generated by

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

S3×M4(2) in GAP, Magma, Sage, TeX

S_3\times M_4(2)
% in TeX

G:=Group("S3xM4(2)");
// GroupNames label

G:=SmallGroup(96,113);
// by ID

G=gap.SmallGroup(96,113);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,188,50,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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