p-group, non-abelian, nilpotent (class 4), monomial, rational
Aliases: D4≀C2, C2≀D4, C24⋊2D4, C42⋊3D4, 2+ 1+4⋊1C22, D42⋊2C2, C2≀C4⋊1C2, (C2×D4)⋊3D4, C22⋊C4⋊1D4, C23⋊C4⋊C22, C2≀C22⋊1C2, D4⋊4D4⋊1C2, C42⋊C4⋊5C2, C2.20C2≀C22, (C2×D4).1C23, C23.13(C2×D4), C4.D4⋊1C22, C22≀C2.3C22, C22.44C22≀C2, C4⋊1D4.55C22, (C2×C4).13(C2×D4), 2-Sylow(S8), SmallGroup(128,928)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4≀C2
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, eae-1=faf=ab=ba, ac=ca, ad=da, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >
Subgroups: 576 in 177 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), D8, SD16, C22×C4, C2×D4, C2×D4, C4○D4, C24, C24, C23⋊C4, C23⋊C4, C4.D4, C4≀C2, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C4⋊1D4, C8⋊C22, C22×D4, 2+ 1+4, C2≀C4, C42⋊C4, D4⋊4D4, C2≀C22, D42, D4≀C2
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, D4≀C2
Character table of D4≀C2
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8 | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 16 | |
| ρ1 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 0 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ16 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C22 |
| ρ17 | 4 | -4 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ19 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C2≀C22 |
| ρ20 | 4 | -4 | 0 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T35
Generators in S8
(1 5)(2 3)(4 7)(6 8)
(1 7)(2 6)(3 8)(4 5)
(2 3)(6 8)
(1 4)(2 3)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)
(1 3)(2 4)(5 6)(7 8)
G:=sub<Sym(8)| (1,5)(2,3)(4,7)(6,8), (1,7)(2,6)(3,8)(4,5), (2,3)(6,8), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8), (1,3)(2,4)(5,6)(7,8)>;
G:=Group( (1,5)(2,3)(4,7)(6,8), (1,7)(2,6)(3,8)(4,5), (2,3)(6,8), (1,4)(2,3)(5,7)(6,8), (1,2)(3,4)(5,6,7,8), (1,3)(2,4)(5,6)(7,8) );
G=PermutationGroup([[(1,5),(2,3),(4,7),(6,8)], [(1,7),(2,6),(3,8),(4,5)], [(2,3),(6,8)], [(1,4),(2,3),(5,7),(6,8)], [(1,2),(3,4),(5,6,7,8)], [(1,3),(2,4),(5,6),(7,8)]])
G:=TransitiveGroup(8,35);
►On 16 points - transitive group 16T376
Generators in S16
(1 11)(2 16)(3 7)(4 14)(5 15)(6 12)(8 10)(9 13)
(1 5)(2 12)(3 9)(4 8)(6 16)(7 13)(10 14)(11 15)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 8)(2 7)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)
G:=sub<Sym(16)| (1,11)(2,16)(3,7)(4,14)(5,15)(6,12)(8,10)(9,13), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)>;
G:=Group( (1,11)(2,16)(3,7)(4,14)(5,15)(6,12)(8,10)(9,13), (1,5)(2,12)(3,9)(4,8)(6,16)(7,13)(10,14)(11,15), (2,16)(4,14)(6,12)(8,10), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13) );
G=PermutationGroup([[(1,11),(2,16),(3,7),(4,14),(5,15),(6,12),(8,10),(9,13)], [(1,5),(2,12),(3,9),(4,8),(6,16),(7,13),(10,14),(11,15)], [(2,16),(4,14),(6,12),(8,10)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(16,376);
►On 16 points - transitive group 16T388
Generators in S16
(1 11)(2 7)(3 13)(4 5)(6 15)(8 9)(10 14)(12 16)
(1 15)(2 12)(3 9)(4 14)(5 10)(6 11)(7 16)(8 13)
(2 7)(4 5)(10 14)(12 16)
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 16)(10 15)(11 14)(12 13)
G:=sub<Sym(16)| (1,11)(2,7)(3,13)(4,5)(6,15)(8,9)(10,14)(12,16), (1,15)(2,12)(3,9)(4,14)(5,10)(6,11)(7,16)(8,13), (2,7)(4,5)(10,14)(12,16), (1,6)(2,7)(3,8)(4,5)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,16)(10,15)(11,14)(12,13)>;
G:=Group( (1,11)(2,7)(3,13)(4,5)(6,15)(8,9)(10,14)(12,16), (1,15)(2,12)(3,9)(4,14)(5,10)(6,11)(7,16)(8,13), (2,7)(4,5)(10,14)(12,16), (1,6)(2,7)(3,8)(4,5)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,16)(10,15)(11,14)(12,13) );
G=PermutationGroup([[(1,11),(2,7),(3,13),(4,5),(6,15),(8,9),(10,14),(12,16)], [(1,15),(2,12),(3,9),(4,14),(5,10),(6,11),(7,16),(8,13)], [(2,7),(4,5),(10,14),(12,16)], [(1,6),(2,7),(3,8),(4,5),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5),(2,8),(3,7),(4,6),(9,16),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(16,388);
►On 16 points - transitive group 16T390
Generators in S16
(2 11)(3 13)(4 9)(5 12)(6 14)(8 16)
(1 10)(2 8)(3 5)(4 9)(6 14)(7 15)(11 16)(12 13)
(2 16)(4 14)(6 9)(8 11)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 16)(14 15)
G:=sub<Sym(16)| (2,11)(3,13)(4,9)(5,12)(6,14)(8,16), (1,10)(2,8)(3,5)(4,9)(6,14)(7,15)(11,16)(12,13), (2,16)(4,14)(6,9)(8,11), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)>;
G:=Group( (2,11)(3,13)(4,9)(5,12)(6,14)(8,16), (1,10)(2,8)(3,5)(4,9)(6,14)(7,15)(11,16)(12,13), (2,16)(4,14)(6,9)(8,11), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15) );
G=PermutationGroup([[(2,11),(3,13),(4,9),(5,12),(6,14),(8,16)], [(1,10),(2,8),(3,5),(4,9),(6,14),(7,15),(11,16),(12,13)], [(2,16),(4,14),(6,9),(8,11)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12),(13,16),(14,15)]])
G:=TransitiveGroup(16,390);
►On 16 points - transitive group 16T391
Generators in S16
(1 3)(2 9)(4 6)(5 12)(7 10)(8 16)(11 14)(13 15)
(1 5)(2 6)(3 12)(4 9)(7 13)(8 14)(10 15)(11 16)
(1 15)(3 13)(5 10)(7 12)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 16)(14 15)
G:=sub<Sym(16)| (1,3)(2,9)(4,6)(5,12)(7,10)(8,16)(11,14)(13,15), (1,5)(2,6)(3,12)(4,9)(7,13)(8,14)(10,15)(11,16), (1,15)(3,13)(5,10)(7,12), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,16)(14,15)>;
G:=Group( (1,3)(2,9)(4,6)(5,12)(7,10)(8,16)(11,14)(13,15), (1,5)(2,6)(3,12)(4,9)(7,13)(8,14)(10,15)(11,16), (1,15)(3,13)(5,10)(7,12), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,16)(14,15) );
G=PermutationGroup([[(1,3),(2,9),(4,6),(5,12),(7,10),(8,16),(11,14),(13,15)], [(1,5),(2,6),(3,12),(4,9),(7,13),(8,14),(10,15),(11,16)], [(1,15),(3,13),(5,10),(7,12)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,9),(6,12),(7,11),(8,10),(13,16),(14,15)]])
G:=TransitiveGroup(16,391);
►On 16 points - transitive group 16T393
Generators in S16
(1 6)(2 7)(3 9)(4 5)(8 16)(10 13)(11 14)(12 15)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)
G:=sub<Sym(16)| (1,6)(2,7)(3,9)(4,5)(8,16)(10,13)(11,14)(12,15), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)>;
G:=Group( (1,6)(2,7)(3,9)(4,5)(8,16)(10,13)(11,14)(12,15), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16) );
G=PermutationGroup([[(1,6),(2,7),(3,9),(4,5),(8,16),(10,13),(11,14),(12,15)], [(2,15),(3,16),(7,12),(8,9)], [(2,15),(4,13),(5,10),(7,12)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5),(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(12,16)]])
G:=TransitiveGroup(16,393);
►On 16 points - transitive group 16T395
Generators in S16
(3 13)(4 9)(6 12)(7 14)
(2 16)(3 6)(4 9)(5 11)(7 14)(12 13)
(1 10)(2 16)(3 12)(4 14)(5 11)(6 13)(7 9)(8 15)
(1 8)(2 5)(3 6)(4 7)(9 14)(10 15)(11 16)(12 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 7)(9 16)(10 15)(11 14)(12 13)
G:=sub<Sym(16)| (3,13)(4,9)(6,12)(7,14), (2,16)(3,6)(4,9)(5,11)(7,14)(12,13), (1,10)(2,16)(3,12)(4,14)(5,11)(6,13)(7,9)(8,15), (1,8)(2,5)(3,6)(4,7)(9,14)(10,15)(11,16)(12,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,16)(10,15)(11,14)(12,13)>;
G:=Group( (3,13)(4,9)(6,12)(7,14), (2,16)(3,6)(4,9)(5,11)(7,14)(12,13), (1,10)(2,16)(3,12)(4,14)(5,11)(6,13)(7,9)(8,15), (1,8)(2,5)(3,6)(4,7)(9,14)(10,15)(11,16)(12,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,16)(10,15)(11,14)(12,13) );
G=PermutationGroup([[(3,13),(4,9),(6,12),(7,14)], [(2,16),(3,6),(4,9),(5,11),(7,14),(12,13)], [(1,10),(2,16),(3,12),(4,14),(5,11),(6,13),(7,9),(8,15)], [(1,8),(2,5),(3,6),(4,7),(9,14),(10,15),(11,16),(12,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,4),(5,7),(9,16),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(16,395);
►On 16 points - transitive group 16T396
Generators in S16
(3 16)(8 9)
(2 15)(3 16)(7 12)(8 9)
(2 15)(4 13)(5 10)(7 12)
(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)
G:=sub<Sym(16)| (3,16)(8,9), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)>;
G:=Group( (3,16)(8,9), (2,15)(3,16)(7,12)(8,9), (2,15)(4,13)(5,10)(7,12), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16) );
G=PermutationGroup([[(3,16),(8,9)], [(2,15),(3,16),(7,12),(8,9)], [(2,15),(4,13),(5,10),(7,12)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,5),(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(12,16)]])
G:=TransitiveGroup(16,396);
►On 16 points - transitive group 16T401
Generators in S16
(2 9)(4 16)(5 11)(7 14)
(1 12)(2 9)(3 15)(4 16)(5 11)(6 10)(7 14)(8 13)
(1 6)(3 8)(10 12)(13 15)
(1 6)(2 5)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 16)(11 15)(12 14)
G:=sub<Sym(16)| (2,9)(4,16)(5,11)(7,14), (1,12)(2,9)(3,15)(4,16)(5,11)(6,10)(7,14)(8,13), (1,6)(3,8)(10,12)(13,15), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,8)(3,5)(4,6)(9,13)(10,16)(11,15)(12,14)>;
G:=Group( (2,9)(4,16)(5,11)(7,14), (1,12)(2,9)(3,15)(4,16)(5,11)(6,10)(7,14)(8,13), (1,6)(3,8)(10,12)(13,15), (1,6)(2,5)(3,8)(4,7)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,7)(2,8)(3,5)(4,6)(9,13)(10,16)(11,15)(12,14) );
G=PermutationGroup([[(2,9),(4,16),(5,11),(7,14)], [(1,12),(2,9),(3,15),(4,16),(5,11),(6,10),(7,14),(8,13)], [(1,6),(3,8),(10,12),(13,15)], [(1,6),(2,5),(3,8),(4,7),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,7),(2,8),(3,5),(4,6),(9,13),(10,16),(11,15),(12,14)]])
G:=TransitiveGroup(16,401);
Polynomial with Galois group D4≀C2 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 8T35 | x8-x7-x6-x5-x3-x2-x+1 | -28·54·892 |
Matrix representation of D4≀C2 ►in GL4(ℤ) generated by
| -1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,Integers())| [-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,0,-1,0,0,-1,0,-1,0,0,0,0,-1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;
D4≀C2 in GAP, Magma, Sage, TeX
D_4\wr C_2
% in TeX
G:=Group("D4wrC2"); // GroupNames label
G:=SmallGroup(128,928);
// by ID
G=gap.SmallGroup(128,928);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,422,297,1971,375,4037]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,e*a*e^-1=f*a*f=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations
Export