p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4○D8, C4○Q16, D8⋊3C2, C4○SD16, Q16⋊3C2, C4.20D4, SD16⋊3C2, C4.4C23, C8.6C22, C22.1D4, D4.2C22, Q8.2C22, (C2×C8)⋊4C2, C4○D4⋊1C2, C2.14(C2×D4), (C2×C4).29C22, SmallGroup(32,42)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4○D8
G = < a,b,c | a4=c2=1, b4=a2, ab=ba, ac=ca, cbc=a2b3 >
Character table of C4○D8
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | |
| 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 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -√-2 | √-2 | -√2 | √2 | complex faithful |
| ρ12 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | √-2 | -√-2 | √2 | -√2 | complex faithful |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | √-2 | -√-2 | -√2 | √2 | complex faithful |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -√-2 | √-2 | √2 | -√2 | complex faithful |
►On 16 points - transitive group 16T44
(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)
G:=sub<Sym(16)| (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)>;
G:=Group( (1,10,5,14)(2,11,6,15)(3,12,7,16)(4,13,8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15) );
G=PermutationGroup([[(1,10,5,14),(2,11,6,15),(3,12,7,16),(4,13,8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15)]])
G:=TransitiveGroup(16,44);
►On 16 points - transitive group 16T47
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)
G:=sub<Sym(16)| (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)>;
G:=Group( (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10) );
G=PermutationGroup([[(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10)]])
G:=TransitiveGroup(16,47);
C4○D8 is a maximal subgroup of
D8.C4 D8⋊2C4 C8○D8 C8.26D4 Q32⋊C2 D8⋊C22 C4.6S4 C32⋊D8⋊5C2 C62.13D4
D8p⋊C2: C4○D16 C16⋊C22 C4○D24 D24⋊C2 D40⋊7C2 Q8.D10 D56⋊7C2 Q8.D14 ...
D4.D2p: D4○D8 D4○SD16 Q8○D8 D8⋊3S3 Q8.7D6 Q8.13D6 D8⋊3D5 SD16⋊3D5 ...
C4○D8 is a maximal quotient of
C23.24D4 C23.25D4 C4×D8 C4×SD16 C4×Q16 D4⋊D4 C8⋊8D4 C8⋊7D4 D4.Q8 Q8.Q8 C23.19D4 C23.20D4 C42.78C22 C8.5Q8 C32⋊D8⋊5C2 C62.13D4
D4.D2p: D4.7D4 D4.2D4 D8⋊3S3 Q8.7D6 Q8.13D6 D8⋊3D5 SD16⋊3D5 D4.8D10 ...
Q8.D2p: Q8.D4 D24⋊C2 Q8.D10 Q8.D14 D88⋊5C2 D104⋊C2 ...
C4p.D4: C8.18D4 C8.12D4 C4○D24 D40⋊7C2 D56⋊7C2 D88⋊7C2 D104⋊7C2 ...
Matrix representation of C4○D8 ►in GL2(𝔽17) generated by
| 4 | 0 |
| 0 | 4 |
| 0 | 6 |
| 14 | 6 |
| 0 | 6 |
| 3 | 0 |
G:=sub<GL(2,GF(17))| [4,0,0,4],[0,14,6,6],[0,3,6,0] >;
C4○D8 in GAP, Magma, Sage, TeX
C_4\circ D_8
% in TeX
G:=Group("C4oD8"); // GroupNames label
G:=SmallGroup(32,42);
// by ID
G=gap.SmallGroup(32,42);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,101,72,483,248,58]);
// Polycyclic
G:=Group<a,b,c|a^4=c^2=1,b^4=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^3>;
// generators/relations
Export
Subgroup lattice of C4○D8 in TeX
Character table of C4○D8 in TeX