p-group, metabelian, nilpotent (class 3), monomial
Aliases: D4.4D4, C8.19D4, Q8.4D4, M4(2).4C22, C8○D4⋊2C2, (C2×D8)⋊8C2, C8⋊C22⋊3C2, C4.59(C2×D4), C8.C4⋊6C2, C4.D4⋊4C2, (C2×C4).9C23, (C2×C8).18C22, C2.24(C4⋊D4), C4○D4.10C22, (C2×D4).20C22, C22.7(C4○D4), SmallGroup(64,153)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D4.4D4
G = < a,b,c,d | a4=b2=d2=1, c4=a2, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c3 >
Character table of D4.4D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | |
| size | 1 | 1 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 16 points - transitive group 16T133
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(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 10)(11 16)(12 15)(13 14)
G:=sub<Sym(16)| (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]])
G:=TransitiveGroup(16,133);
D4.4D4 is a maximal subgroup of
M4(2).10C23 D8⋊11D4 D8○D8 C8.3S4
M4(2).D2p: Q16.10D4 Q16.D4 D4.3D8 D4.5D8 M4(2).37D4 D12.3D4 C24.19D4 Q8.9D12 ...
D4p.D4: D8○SD16 C24.23D4 C40.23D4 C56.23D4 ...
D4.4D4 is a maximal quotient of
M4(2).D2p: M4(2).48D4 M4(2).32D4 M4(2).4D4 M4(2).10D4 M4(2).12D4 D12.3D4 C24.19D4 Q8.9D12 ...
(C2×C8).D2p: C8⋊7D8 C8⋊13SD16 D4.1Q16 Q8.1Q16 D4.2SD16 Q8.2SD16 C8⋊2D8 C8⋊2SD16 ...
Matrix representation of D4.4D4 ►in GL4(𝔽7) generated by
| 0 | 6 | 5 | 1 |
| 3 | 0 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 1 | 6 | 3 | 1 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 6 | 6 | 4 | 3 |
| 2 | 5 | 2 | 3 |
| 5 | 0 | 2 | 6 |
| 6 | 5 | 5 | 6 |
| 6 | 1 | 1 | 2 |
| 2 | 2 | 6 | 2 |
| 1 | 4 | 1 | 6 |
| 0 | 2 | 5 | 4 |
| 0 | 6 | 2 | 1 |
| 0 | 4 | 2 | 2 |
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[0,2,6,2,4,0,6,5,0,0,4,2,0,0,3,3],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[1,0,0,0,4,2,6,4,1,5,2,2,6,4,1,2] >;
D4.4D4 in GAP, Magma, Sage, TeX
D_4._4D_4
% in TeX
G:=Group("D4.4D4"); // GroupNames label
G:=SmallGroup(64,153);
// by ID
G=gap.SmallGroup(64,153);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,121,247,362,963,117,1444,376,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^4=a^2,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=a^2*c^3>;
// generators/relations
Export
Subgroup lattice of D4.4D4 in TeX
Character table of D4.4D4 in TeX