metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.11D12, C12.46D4, M4(2)⋊3S3, (C2×C4).1D6, (C22×S3).C4, C2.9(D6⋊C4), (C2×D12).6C2, C3⋊1(C4.D4), C4.Dic3⋊2C2, C22.4(C4×S3), C4.21(C3⋊D4), C6.8(C22⋊C4), (C3×M4(2))⋊7C2, (C2×C12).13C22, (C2×C6).2(C2×C4), SmallGroup(96,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊S3
G = < a,b,c,d | a8=b2=c3=d2=1, bab=a5, ac=ca, dad=ab, bc=cb, bd=db, dcd=c-1 >
2C2
12C2
12C2
6C22
6C22
12C22
12C22
2C6
4S3
4S3
2C8
3C23
3C23
6C8
6D4
6D4
2D6
2D6
4D6
4D6
2C24
2D12
2D12
Character table of M4(2)⋊S3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ15 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | -2i | 2i | 0 | 0 | 1 | 1 | 1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | 2i | -2i | 0 | 0 | 1 | 1 | 1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 24 points - transitive group 24T105
Generators in S24
(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)(9 13)(11 15)(17 21)(19 23)
(1 10 22)(2 11 23)(3 12 24)(4 13 17)(5 14 18)(6 15 19)(7 16 20)(8 9 21)
(2 6)(3 7)(9 21)(10 22)(11 19)(12 20)(13 17)(14 18)(15 23)(16 24)
G:=sub<Sym(24)| (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)(9,13)(11,15)(17,21)(19,23), (1,10,22)(2,11,23)(3,12,24)(4,13,17)(5,14,18)(6,15,19)(7,16,20)(8,9,21), (2,6)(3,7)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)(15,23)(16,24)>;
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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23), (1,10,22)(2,11,23)(3,12,24)(4,13,17)(5,14,18)(6,15,19)(7,16,20)(8,9,21), (2,6)(3,7)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)(15,23)(16,24) );
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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23)], [(1,10,22),(2,11,23),(3,12,24),(4,13,17),(5,14,18),(6,15,19),(7,16,20),(8,9,21)], [(2,6),(3,7),(9,21),(10,22),(11,19),(12,20),(13,17),(14,18),(15,23),(16,24)])
G:=TransitiveGroup(24,105);
Matrix representation of M4(2)⋊S3 ►in GL6(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 68 | 21 | 3 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 43 | 23 | 7 | 5 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 17 | 52 | 0 | 72 |
| 36 | 28 | 0 | 0 | 0 | 0 |
| 28 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 68 | 21 | 3 |
| 0 | 0 | 22 | 7 | 48 | 52 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,43,0,0,0,68,72,23,0,0,1,21,0,7,0,0,0,3,0,5],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,17,0,0,0,1,0,52,0,0,0,0,72,0,0,0,0,0,0,72],[36,28,0,0,0,0,28,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,11,22,0,0,1,0,68,7,0,0,0,0,21,48,0,0,0,0,3,52] >;
M4(2)⋊S3 in GAP, Magma, Sage, TeX
M_{4(2})\rtimes S_3 % in TeX
G:=Group("M4(2):S3"); // GroupNames label
G:=SmallGroup(96,30);
// by ID
G=gap.SmallGroup(96,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,121,31,362,86,297,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^5,a*c=c*a,d*a*d=a*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations