non-abelian, soluble, monomial
Aliases: D4⋊S4, A4⋊2D8, C4⋊S4⋊2C2, A4⋊C8⋊2C2, (D4×A4)⋊1C2, C4.2(C2×S4), C22⋊(D4⋊S3), (C2×A4).8D4, (C22×D4)⋊1S3, (C22×C4).2D6, (C4×A4).2C22, C2.5(A4⋊D4), C23.18(C3⋊D4), SmallGroup(192,974)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊S4
G = < a,b,c,d,e,f | a4=b2=c2=d2=e3=f2=1, bab=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf=ab, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 454 in 87 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, D4, D4, C23, C23, C12, A4, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C3⋊C8, D12, C3×D4, S4, C2×A4, C2×A4, C22⋊C8, D4⋊C4, C4⋊D4, C2×D8, C22×D4, D4⋊S3, C4×A4, C2×S4, C22×A4, C22⋊D8, A4⋊C8, C4⋊S4, D4×A4, D4⋊S4
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊D4, S4, D4⋊S3, C2×S4, A4⋊D4, D4⋊S4
Character table of D4⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12 | |
| size | 1 | 1 | 3 | 3 | 4 | 12 | 24 | 8 | 2 | 6 | 24 | 8 | 16 | 16 | 12 | 12 | 12 | 12 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 2 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | orthogonal lifted from D8 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -2 | -2 | 0 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | complex lifted from C3⋊D4 |
| ρ12 | 3 | 3 | -1 | -1 | -3 | 1 | -1 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | orthogonal lifted from C2×S4 |
| ρ13 | 3 | 3 | -1 | -1 | 3 | -1 | 1 | 0 | 3 | -1 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | 3 | -1 | 1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | -1 | -1 | -3 | 1 | 1 | 0 | 3 | -1 | -1 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ17 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -6 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
| ρ18 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | 0 | orthogonal faithful |
| ρ19 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | 0 | orthogonal faithful |
►On 24 points - transitive group 24T327
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 24)(2 23)(3 22)(4 21)(5 15)(6 14)(7 13)(8 16)(9 20)(10 19)(11 18)(12 17)
(1 3)(2 4)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 11 16)(2 12 13)(3 9 14)(4 10 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(2 4)(5 20)(6 19)(7 18)(8 17)(9 14)(10 13)(11 16)(12 15)(21 22)(23 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), (1,24)(2,23)(3,22)(4,21)(5,15)(6,14)(7,13)(8,16)(9,20)(10,19)(11,18)(12,17), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,16)(2,12,13)(3,9,14)(4,10,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (2,4)(5,20)(6,19)(7,18)(8,17)(9,14)(10,13)(11,16)(12,15)(21,22)(23,24)>;
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), (1,24)(2,23)(3,22)(4,21)(5,15)(6,14)(7,13)(8,16)(9,20)(10,19)(11,18)(12,17), (1,3)(2,4)(9,11)(10,12)(17,19)(18,20)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,11,16)(2,12,13)(3,9,14)(4,10,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (2,4)(5,20)(6,19)(7,18)(8,17)(9,14)(10,13)(11,16)(12,15)(21,22)(23,24) );
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)], [(1,24),(2,23),(3,22),(4,21),(5,15),(6,14),(7,13),(8,16),(9,20),(10,19),(11,18),(12,17)], [(1,3),(2,4),(9,11),(10,12),(17,19),(18,20),(21,23),(22,24)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,11,16),(2,12,13),(3,9,14),(4,10,15),(5,21,19),(6,22,20),(7,23,17),(8,24,18)], [(2,4),(5,20),(6,19),(7,18),(8,17),(9,14),(10,13),(11,16),(12,15),(21,22),(23,24)]])
G:=TransitiveGroup(24,327);
►On 24 points - transitive group 24T330
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)(5 7)(9 12)(10 11)(13 14)(15 16)(17 19)(21 23)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 9 16)(2 10 13)(3 11 14)(4 12 15)(5 19 21)(6 20 22)(7 17 23)(8 18 24)
(1 21)(2 24)(3 23)(4 22)(5 16)(6 15)(7 14)(8 13)(9 19)(10 18)(11 17)(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)(5,7)(9,12)(10,11)(13,14)(15,16)(17,19)(21,23), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,9,16)(2,10,13)(3,11,14)(4,12,15)(5,19,21)(6,20,22)(7,17,23)(8,18,24), (1,21)(2,24)(3,23)(4,22)(5,16)(6,15)(7,14)(8,13)(9,19)(10,18)(11,17)(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)(5,7)(9,12)(10,11)(13,14)(15,16)(17,19)(21,23), (9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (1,9,16)(2,10,13)(3,11,14)(4,12,15)(5,19,21)(6,20,22)(7,17,23)(8,18,24), (1,21)(2,24)(3,23)(4,22)(5,16)(6,15)(7,14)(8,13)(9,19)(10,18)(11,17)(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),(5,7),(9,12),(10,11),(13,14),(15,16),(17,19),(21,23)], [(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(13,15),(14,16),(21,23),(22,24)], [(1,9,16),(2,10,13),(3,11,14),(4,12,15),(5,19,21),(6,20,22),(7,17,23),(8,18,24)], [(1,21),(2,24),(3,23),(4,22),(5,16),(6,15),(7,14),(8,13),(9,19),(10,18),(11,17),(12,20)]])
G:=TransitiveGroup(24,330);
Matrix representation of D4⋊S4 ►in GL5(𝔽73)
| 33 | 32 | 0 | 0 | 0 |
| 7 | 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 39 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(73))| [33,7,0,0,0,32,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,72,72,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,1,0,0,0,0,0,0,1],[1,39,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
D4⋊S4 in GAP, Magma, Sage, TeX
D_4\rtimes S_4
% in TeX
G:=Group("D4:S4"); // GroupNames label
G:=SmallGroup(192,974);
// by ID
G=gap.SmallGroup(192,974);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,85,254,135,58,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=c^2=d^2=e^3=f^2=1,b*a*b=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
Export