direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3xM4(2), C8:6D6, C24:7C22, Dic3oM4(2), C12.38C23, (S3xC8):7C2, C8:S3:5C2, (C4xS3).1C4, C4.15(C4xS3), C3:C8:11C22, D6.6(C2xC4), (C2xC4).45D6, C3:2(C2xM4(2)), C12.12(C2xC4), C4.Dic3:5C2, C22.7(C4xS3), (C3xM4(2)):5C2, (C22xS3).4C4, C6.15(C22xC4), C4.38(C22xS3), (C2xDic3).6C4, Dic3.7(C2xC4), (C4xS3).18C22, (C2xC12).25C22, (S3xC2xC4).4C2, C2.16(S3xC2xC4), (C2xC6).5(C2xC4), SmallGroup(96,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3xM4(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, C2xC4, C2xC4, C23, Dic3, C12, D6, D6, C2xC6, C2xC8, M4(2), M4(2), C22xC4, C3:C8, C24, C4xS3, C2xDic3, C2xC12, C22xS3, C2xM4(2), S3xC8, C8:S3, C4.Dic3, C3xM4(2), S3xC2xC4, S3xM4(2)
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, M4(2), C22xC4, C4xS3, C22xS3, C2xM4(2), S3xC2xC4, S3xM4(2)
Character table of S3xM4(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 C4xS3 |
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 C4xS3 |
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 C4xS3 |
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 C4xS3 |
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 |
(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);
S3xM4(2) is a maximal subgroup of
M4(2).19D6 M4(2).21D6 C42:3D6 M4(2).25D6 M4(2):26D6 M4(2):28D6 D8:4D6 D24:C22 C24:D6 C3:C8:20D6 C40:D6 D15:4M4(2) D15:M4(2) D15:2M4(2)
S3xM4(2) is a maximal quotient of
C24:Q8 C42.182D6 C8:9D12 Dic3:5M4(2) Dic3.5M4(2) Dic3.M4(2) D6:M4(2) D6:2M4(2) Dic3:M4(2) C42.27D6 C42.200D6 C42.202D6 D6:3M4(2) C12:M4(2) Dic3:4M4(2) D6:6M4(2) C24:D4 C24:21D4 C24:D6 C3:C8:20D6 C40:D6 D15:4M4(2) D15:M4(2) D15:2M4(2)
Matrix representation of S3xM4(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] >;
S3xM4(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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