metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2S3, D4○Dic3, C4.5D6, Dic6⋊3C2, C6.6C23, C22.1D6, C12.5C22, D6.2C22, Dic3.4C22, (C4×S3)⋊2C2, (C3×D4)⋊3C2, C3⋊2(C4○D4), C3⋊D4⋊2C2, (C2×C6).C22, (C2×Dic3)⋊3C2, C2.7(C22×S3), SmallGroup(48,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2S3
G = < a,b,c,d | a4=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Character table of D4⋊2S3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 12 | |
| size | 1 | 1 | 2 | 2 | 6 | 2 | 2 | 3 | 3 | 6 | 6 | 2 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 2 | 0 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T18
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
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), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20)>;
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), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,24)(22,23), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20) );
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)], [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12),(13,14),(15,16),(17,20),(18,19),(21,24),(22,23)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)]])
G:=TransitiveGroup(24,18);
►On 24 points - transitive group 24T23
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 10)(2 9)(3 12)(4 11)(5 21)(6 24)(7 23)(8 22)(13 20)(14 19)(15 18)(16 17)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 13)(10 24 14)(11 21 15)(12 22 16)
(5 18)(6 19)(7 20)(8 17)(9 11)(10 12)(13 21)(14 22)(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), (1,10)(2,9)(3,12)(4,11)(5,21)(6,24)(7,23)(8,22)(13,20)(14,19)(15,18)(16,17), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,13)(10,24,14)(11,21,15)(12,22,16), (5,18)(6,19)(7,20)(8,17)(9,11)(10,12)(13,21)(14,22)(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), (1,10)(2,9)(3,12)(4,11)(5,21)(6,24)(7,23)(8,22)(13,20)(14,19)(15,18)(16,17), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,13)(10,24,14)(11,21,15)(12,22,16), (5,18)(6,19)(7,20)(8,17)(9,11)(10,12)(13,21)(14,22)(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)], [(1,10),(2,9),(3,12),(4,11),(5,21),(6,24),(7,23),(8,22),(13,20),(14,19),(15,18),(16,17)], [(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,23,13),(10,24,14),(11,21,15),(12,22,16)], [(5,18),(6,19),(7,20),(8,17),(9,11),(10,12),(13,21),(14,22),(15,23),(16,24)]])
G:=TransitiveGroup(24,23);
D4⋊2S3 is a maximal subgroup of
D8⋊S3 D8⋊3S3 D4.D6 Q8.7D6 D4⋊6D6 S3×C4○D4 Q8○D12 D4⋊2D9 D12⋊5S3 D12⋊S3 D6.3D6 D6.4D6 C12.D6 D4⋊2S4 D4.5S4 D20⋊5S3 D20⋊S3 Dic5.D6 C30.C23 D4⋊2D15 D28⋊5S3 D28⋊S3 Dic7.D6 C42.C23 D4⋊2D21 C22.S5 D4.A5
D4⋊2S3 is a maximal quotient of
C23.16D6 Dic3.D4 C23.8D6 Dic3⋊4D4 C23.9D6 C23.11D6 C23.21D6 Dic6⋊C4 Dic3.Q8 C4.Dic6 C4⋊C4⋊7S3 C4.D12 C4⋊C4⋊S3 D4×Dic3 C23.23D6 C23.12D6 D6⋊3D4 C23.14D6 D4⋊2D9 D12⋊5S3 D12⋊S3 D6.3D6 D6.4D6 C12.D6 D4⋊2S4 D20⋊5S3 D20⋊S3 Dic5.D6 C30.C23 D4⋊2D15 D28⋊5S3 D28⋊S3 Dic7.D6 C42.C23 D4⋊2D21
Matrix representation of D4⋊2S3 ►in GL4(𝔽5) generated by
| 0 | 1 | 3 | 0 |
| 1 | 3 | 2 | 3 |
| 1 | 4 | 1 | 4 |
| 0 | 2 | 3 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 3 | 4 | 0 | 4 |
| 3 | 4 | 4 | 0 |
| 3 | 0 | 0 | 4 |
| 2 | 4 | 1 | 1 |
| 1 | 4 | 0 | 1 |
| 3 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 4 |
| 0 | 1 | 0 | 4 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,1,1,0,1,3,4,2,3,2,1,3,0,3,4,1],[1,0,3,3,1,4,4,4,0,0,0,4,0,0,4,0],[3,2,1,3,0,4,4,0,0,1,0,0,4,1,1,1],[1,0,0,0,0,0,1,0,0,1,0,0,1,4,4,4] >;
D4⋊2S3 in GAP, Magma, Sage, TeX
D_4\rtimes_2S_3
% in TeX
G:=Group("D4:2S3"); // GroupNames label
G:=SmallGroup(48,39);
// by ID
G=gap.SmallGroup(48,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,46,182,97,804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of D4⋊2S3 in TeX
Character table of D4⋊2S3 in TeX