p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8oD4, C8oQ8, D4.C4, Q8.C4, C8oM4(2), C8.7C22, M4(2):5C2, C4.12C23, (C2xC8):7C2, C8o(C4oD4), C4.5(C2xC4), C4oD4.3C2, C22.1(C2xC4), C2.7(C22xC4), (C2xC4).25C22, SmallGroup(32,38)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8oD4
G = < a,b,c | a8=c2=1, b2=a4, ab=ba, ac=ca, cbc=a4b >
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C8oD4
Character table of C8oD4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | i | i | -i | i | -i | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | i | i | -i | -i | -i | i | i | i | -i | -i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ15 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | -i | -i | i | -i | i | i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -i | -i | i | i | i | -i | -i | -i | i | i | linear of order 4 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2ζ85 | 2ζ8 | 2ζ83 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2ζ83 | 2ζ87 | 2ζ85 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2ζ8 | 2ζ85 | 2ζ87 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2ζ87 | 2ζ83 | 2ζ8 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)
(1 9)(2 10)(3 11)(4 12)(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), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16), (1,9)(2,10)(3,11)(4,12)(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), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16), (1,9)(2,10)(3,11)(4,12)(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)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)]])
G:=TransitiveGroup(16,16);
C8oD4 is a maximal subgroup of
C8.A4 D4.F5 Q8.F5 C62.(C2xC4) C12:S3.C4 Dic26.C4 D52.C4
C8.D2p: D4.C8 C8oD8 C8.26D4 D4.3D4 D4.4D4 D4.5D4 C8oD12 D12.C4 ...
C4p.C23: D4oC16 Q8oM4(2) D4oD8 D4oSD16 Q8oD8 D4.Dic3 D4.Dic5 Q8.Dic7 ...
C8oD4 is a maximal quotient of
(C22xC8):C2 C42.6C22 C42.7C22 C8xD4 C8:9D4 C8:6D4 C8xQ8 C8:4Q8 C62.(C2xC4) C12:S3.C4
C4p.(C2xC4): C8o2M4(2) C8oD12 D12.C4 D4.Dic3 D20.3C4 D20.2C4 D4.Dic5 D4.F5 ...
Matrix representation of C8oD4 ►in GL2(F17) generated by
| 8 | 0 |
| 0 | 8 |
| 0 | 1 |
| 16 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(17))| [8,0,0,8],[0,16,1,0],[0,1,1,0] >;
C8oD4 in GAP, Magma, Sage, TeX
C_8\circ D_4
% in TeX
G:=Group("C8oD4"); // GroupNames label
G:=SmallGroup(32,38);
// by ID
G=gap.SmallGroup(32,38);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,40,157,58]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^2=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b>;
// generators/relations
Export
Subgroup lattice of C8oD4 in TeX
Character table of C8oD4 in TeX