p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: 2+ 1+4, D4○D4, Q8○Q8, D4⋊4C22, C2.4C24, C4.9C23, Q8⋊4C22, C23⋊2C22, C22.2C23, C4○D4⋊3C2, (C2×D4)⋊6C2, (C2×C4)⋊2C22, 2-Sylow(Omega+(4,3)), SmallGroup(32,49)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 2+ 1+4
G = < a,b,c,d | a4=b2=d2=1, c2=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c >
Subgroups: 110 in 83 conjugacy classes, 68 normal (3 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, Q8, C23, C2×D4, C4○D4, 2+ 1+4
Quotients: C1, C2, C22, C23, C24, 2+ 1+4
Character table of 2+ 1+4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 8 points - transitive group 8T22
(1 2 3 4)(5 6 7 8)
(1 4)(2 3)(5 8)(6 7)
(1 5 3 7)(2 6 4 8)
(1 7)(2 8)(3 5)(4 6)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7), (1,5,3,7)(2,6,4,8), (1,7)(2,8)(3,5)(4,6)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,4)(2,3)(5,8)(6,7), (1,5,3,7)(2,6,4,8), (1,7)(2,8)(3,5)(4,6) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,4),(2,3),(5,8),(6,7)], [(1,5,3,7),(2,6,4,8)], [(1,7),(2,8),(3,5),(4,6)]])
G:=TransitiveGroup(8,22);
►On 16 points - transitive group 16T23
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14)(2 13)(3 16)(4 15)(5 10)(6 9)(7 12)(8 11)
(1 6 3 8)(2 7 4 5)(9 16 11 14)(10 13 12 15)
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14)(2,13)(3,16)(4,15)(5,10)(6,9)(7,12)(8,11), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14)(2,13)(3,16)(4,15)(5,10)(6,9)(7,12)(8,11), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14),(2,13),(3,16),(4,15),(5,10),(6,9),(7,12),(8,11)], [(1,6,3,8),(2,7,4,5),(9,16,11,14),(10,13,12,15)], [(1,8),(2,5),(3,6),(4,7),(9,16),(10,13),(11,14),(12,15)]])
G:=TransitiveGroup(16,23);
2+ 1+4 is a maximal subgroup of
D4.9D4 C2≀C22 C23.7D4 D4○D8 D4○SD16 C2.C25 Q8.A4 C23⋊A4
D4⋊D2p: D4⋊4D4 D4⋊6D6 D4○D12 D4⋊6D10 D4⋊8D10 D4⋊6D14 D4⋊8D14 D4⋊6D22 ...
2+ 1+4 is a maximal quotient of
C22.11C24 C23.33C23 C23⋊3D4 C22.29C24 C22.31C24 C22.32C24 C22.33C24 C22.34C24 C22.36C24 C23⋊2Q8 C23.41C23 D42 Q8⋊6D4 C22.45C24 C22.47C24 D4⋊3Q8 C22.49C24 Q82 C22.53C24 C22.54C24 C24⋊C22 C22.56C24 C22.57C24
D4⋊D2p: D4⋊5D4 D4⋊6D6 D4○D12 D4⋊6D10 D4⋊8D10 D4⋊6D14 D4⋊8D14 D4⋊6D22 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T22 | x8-16x6-12x5+73x4+96x3-56x2-124x-41 | 216·32·54·72·312 |
Matrix representation of 2+ 1+4 ►in GL4(ℤ) generated by
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 |
G:=sub<GL(4,Integers())| [0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0] >;
2+ 1+4 in GAP, Magma, Sage, TeX
2_+^{1+4} % in TeX
G:=Group("ES+(2,2)"); // GroupNames label
G:=SmallGroup(32,49);
// by ID
G=gap.SmallGroup(32,49);
# by ID
G:=PCGroup([5,-2,2,2,2,-2,181,157,483]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c>;
// generators/relations
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