metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊S3, C3⋊2D8, C4.1D6, C6.7D4, D12⋊2C2, C12.1C22, C3⋊C8⋊1C2, (C3×D4)⋊1C2, C2.4(C3⋊D4), SmallGroup(48,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊S3
G = < a,b,c,d | a4=b2=c3=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Character table of D4⋊S3
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 8A | 8B | 12 | |
| size | 1 | 1 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -√2 | √2 | 0 | orthogonal lifted from D8 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | √2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | -√-3 | √-3 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -2 | -1 | √-3 | -√-3 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ12 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 24 points - transitive group 24T37
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)
(1 21)(2 24)(3 23)(4 22)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)
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), (1,21)(2,24)(3,23)(4,22)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (2,4)(5,9)(6,12)(7,11)(8,10)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)>;
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), (1,21)(2,24)(3,23)(4,22)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (2,4)(5,9)(6,12)(7,11)(8,10)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23) );
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)], [(1,21),(2,24),(3,23),(4,22),(5,19),(6,18),(7,17),(8,20),(9,15),(10,14),(11,13),(12,16)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(2,4),(5,9),(6,12),(7,11),(8,10),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23)]])
G:=TransitiveGroup(24,37);
►On 24 points - transitive group 24T43
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)
(1 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(22 24)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 22)(2 21)(3 24)(4 23)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)
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)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)>;
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)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13) );
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),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(22,24)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(1,22),(2,21),(3,24),(4,23),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(24,43);
D4⋊S3 is a maximal subgroup of
S3×D8 D8⋊S3 Q8⋊3D6 Q8.7D6 D12⋊6C22 D4⋊D6 Q8.13D6 D4⋊D9 C32⋊2D8 C3⋊D24 C32⋊7D8 D4⋊S4 D4.3S4 C15⋊D8 C3⋊D40 D4⋊D15 C21⋊D8 C3⋊D56 D4⋊D21 C33⋊D8
D4⋊S3 is a maximal quotient of
C6.Q16 C6.D8 C3⋊D16 D8.S3 C8.6D6 C3⋊Q32 D4⋊Dic3 D4⋊D9 C32⋊2D8 C3⋊D24 C32⋊7D8 D4⋊S4 C15⋊D8 C3⋊D40 D4⋊D15 C21⋊D8 C3⋊D56 D4⋊D21 C33⋊D8
Matrix representation of D4⋊S3 ►in GL4(𝔽5) generated by
| 0 | 0 | 2 | 4 |
| 3 | 4 | 4 | 2 |
| 2 | 1 | 1 | 0 |
| 0 | 2 | 2 | 0 |
| 2 | 1 | 1 | 3 |
| 0 | 0 | 3 | 1 |
| 1 | 0 | 0 | 4 |
| 2 | 1 | 0 | 3 |
| 0 | 3 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 0 | 4 | 4 | 0 |
| 1 | 3 | 3 | 4 |
| 1 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,3,2,0,0,4,1,2,2,4,1,2,4,2,0,0],[2,0,1,2,1,0,0,1,1,3,0,0,3,1,4,3],[0,0,0,1,3,0,4,3,0,1,4,3,4,0,0,4],[1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4] >;
D4⋊S3 in GAP, Magma, Sage, TeX
D_4\rtimes S_3
% in TeX
G:=Group("D4:S3"); // GroupNames label
G:=SmallGroup(48,15);
// by ID
G=gap.SmallGroup(48,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,61,182,97,42,804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^3=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of D4⋊S3 in TeX
Character table of D4⋊S3 in TeX