p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4○D4, C4○Q8, D4⋊2C2, Q8⋊2C2, C4.5C22, C2.3C23, C22.C22, (C2×C4)⋊3C2, Pauli group, SmallGroup(16,13)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4○D4
G = < a,b,c | a4=c2=1, b2=a2, ab=ba, ac=ca, cbc=a2b >
Character table of C4○D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | linear of order 2 |
| ρ6 | 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 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex faithful |
| ρ10 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex faithful |
►On 8 points - transitive group 8T11
(1 2 3 4)(5 6 7 8)
(1 4 3 2)(5 6 7 8)
(1 6)(2 7)(3 8)(4 5)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,4,3,2)(5,6,7,8), (1,6)(2,7)(3,8)(4,5)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,4,3,2)(5,6,7,8), (1,6)(2,7)(3,8)(4,5) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,4,3,2),(5,6,7,8)], [(1,6),(2,7),(3,8),(4,5)]])
G:=TransitiveGroup(8,11);
►Regular action on 16 points - transitive group 16T11
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 3 15)(2 14 4 16)(5 11 7 9)(6 12 8 10)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,3,15)(2,14,4,16)(5,11,7,9)(6,12,8,10), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,3,15)(2,14,4,16)(5,11,7,9)(6,12,8,10), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,13,3,15),(2,14,4,16),(5,11,7,9),(6,12,8,10)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)]])
G:=TransitiveGroup(16,11);
C4○D4 is a maximal subgroup of
C8○D4 2+ 1+4 2- 1+4 C4.A4 D5≀C2⋊C2
D4p⋊C2: C4○D8 C8⋊C22 C4○D12 Q8⋊3S3 C4○D20 Q8⋊2D5 C4○D28 Q8⋊2D7 ...
C4.D2p: C4≀C2 C8.C22 D4⋊2S3 D4⋊2D5 D4⋊2D7 D4⋊2D11 D4⋊2D13 D4⋊2D17 ...
C4○D4 is a maximal quotient of
C42⋊C2 C4×D4 C4×Q8 C4⋊D4 C22.D4 C42.C2 C42⋊2C2 D5≀C2⋊C2
C4.D2p: C22⋊Q8 C4.4D4 C4○D12 D4⋊2S3 Q8⋊3S3 C4○D20 D4⋊2D5 Q8⋊2D5 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T11 | x8-2x7-7x6+16x5+4x4-18x3+2x2+4x-1 | 212·54·132 |
Matrix representation of C4○D4 ►in GL2(𝔽5) generated by
| 2 | 0 |
| 0 | 2 |
| 1 | 3 |
| 1 | 4 |
| 1 | 3 |
| 0 | 4 |
G:=sub<GL(2,GF(5))| [2,0,0,2],[1,1,3,4],[1,0,3,4] >;
C4○D4 in GAP, Magma, Sage, TeX
C_4\circ D_4
% in TeX
G:=Group("C4oD4"); // GroupNames label
G:=SmallGroup(16,13);
// by ID
G=gap.SmallGroup(16,13);
# by ID
G:=PCGroup([4,-2,2,2,-2,81,34]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^2=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b>;
// generators/relations
Export
Subgroup lattice of C4○D4 in TeX
Character table of C4○D4 in TeX