p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4⋊2D4, C22⋊1D4, C23.2C22, C22.11C23, C4⋊C4⋊2C2, (C2×D4)⋊2C2, C2.5(C2×D4), C22⋊C4⋊3C2, (C22×C4)⋊4C2, C2.4(C4○D4), (C2×C4).20C22, SmallGroup(32,28)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊D4
G = < a,b,c | a4=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C4⋊D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 16 points - transitive group 16T34
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 15 9)(2 5 16 12)(3 8 13 11)(4 7 14 10)
(1 4)(2 3)(5 11)(6 10)(7 9)(8 12)(13 16)(14 15)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,15,9)(2,5,16,12)(3,8,13,11)(4,7,14,10), (1,4)(2,3)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,15,9)(2,5,16,12)(3,8,13,11)(4,7,14,10), (1,4)(2,3)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6,15,9),(2,5,16,12),(3,8,13,11),(4,7,14,10)], [(1,4),(2,3),(5,11),(6,10),(7,9),(8,12),(13,16),(14,15)]])
G:=TransitiveGroup(16,34);
►On 16 points - transitive group 16T43
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 9 14)(2 5 10 13)(3 8 11 16)(4 7 12 15)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,9,14)(2,5,10,13)(3,8,11,16)(4,7,12,15), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,9,14)(2,5,10,13)(3,8,11,16)(4,7,12,15), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6,9,14),(2,5,10,13),(3,8,11,16),(4,7,12,15)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]])
G:=TransitiveGroup(16,43);
C4⋊D4 is a maximal subgroup of
C22.19C24 C23.36C23 C22.26C24 C23⋊3D4 C22.29C24 C22.31C24 C22.32C24 C22.33C24 C22.34C24 C22.36C24 D42 Q8⋊5D4 Q8⋊6D4 C22.47C24 C22.49C24 C22.54C24 C22.56C24 S32⋊D4 C62⋊D4
C4p⋊D4: C8⋊8D4 C8⋊7D4 C8⋊D4 C8⋊2D4 C12⋊D4 C12⋊7D4 D6⋊3D4 C4⋊D20 ...
D2p⋊D4: C22⋊D8 D4⋊D4 D4⋊5D4 D4⋊6D4 Dic3⋊D4 D10⋊D4 D14⋊D4 D22⋊D4 ...
C23.D2p: C22.SD16 Q8⋊D4 C22.D8 C23.46D4 C23.19D4 C23.14D6 C4⋊S4 Dic5⋊D4 ...
C4⋊D4 is a maximal quotient of
C24.C22 C23.65C23 C24.3C22 C23⋊2D4 C23.Q8 C23.81C23 C4⋊SD16 D4.D4 C4⋊2Q16 D4.2D4 Q8.D4 D4.3D4 D4.4D4 D4.5D4 S32⋊D4 C62⋊D4
C4p⋊D4: C8⋊8D4 C8⋊7D4 C8⋊D4 C8⋊2D4 C12⋊D4 C12⋊7D4 D6⋊3D4 C4⋊D20 ...
D2p⋊D4: C4⋊D8 Dic3⋊D4 D10⋊D4 D14⋊D4 D22⋊D4 D26⋊D4 ...
C23.D2p: C23.7Q8 C23.23D4 C23.10D4 C23.11D4 C8.18D4 C8.D4 C23.14D6 Dic5⋊D4 ...
Matrix representation of C4⋊D4 ►in GL4(𝔽5) generated by
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 2 |
| 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 4 |
G:=sub<GL(4,GF(5))| [0,4,0,0,1,0,0,0,0,0,2,0,0,0,1,3],[1,0,0,0,0,4,0,0,0,0,4,4,0,0,2,1],[1,0,0,0,0,4,0,0,0,0,1,1,0,0,0,4] >;
C4⋊D4 in GAP, Magma, Sage, TeX
C_4\rtimes D_4
% in TeX
G:=Group("C4:D4"); // GroupNames label
G:=SmallGroup(32,28);
// by ID
G=gap.SmallGroup(32,28);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,101,46,302]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C4⋊D4 in TeX
Character table of C4⋊D4 in TeX