p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8.1C4, C22.Q8, C4.19D4, M4(2).2C2, C4.8(C2×C4), (C2×C8).5C2, C2.5(C4⋊C4), (C2×C4).19C22, SmallGroup(32,15)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.C4
G = < a,b | a8=1, b4=a4, bab-1=a-1 >
Character table of C8.C4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | -2 | 0 | 2i | -2i | 0 | -√2 | √-2 | √2 | -√-2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ12 | 2 | -2 | 0 | 2i | -2i | 0 | √2 | -√-2 | -√2 | √-2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 2 | -2 | 0 | -2i | 2i | 0 | -√2 | -√-2 | √2 | √-2 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 2 | -2 | 0 | -2i | 2i | 0 | √2 | √-2 | -√2 | -√-2 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T49
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 9 3 15 5 13 7 11)(2 16 4 14 6 12 8 10)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9,3,15,5,13,7,11)(2,16,4,14,6,12,8,10)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,9,3,15,5,13,7,11)(2,16,4,14,6,12,8,10) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,9,3,15,5,13,7,11),(2,16,4,14,6,12,8,10)]])
G:=TransitiveGroup(16,49);
C8.C4 is a maximal subgroup of
C8.Q8 C8.4Q8 M4(2).C4 C40.C4 D10.Q8 C4.19S3≀C2 C62.2Q8 (C3×C24).C4 C8.(C32⋊C4) C104.C4 C104.1C4
C4p.D4: D8.C4 M5(2)⋊C2 C8.17D4 C8○D8 C8.26D4 D4.3D4 D4.4D4 D4.5D4 ...
C8.C4 is a maximal quotient of
C4.C42 C40.C4 D10.Q8 C4.19S3≀C2 C62.2Q8 (C3×C24).C4 C8.(C32⋊C4) C104.C4 C104.1C4
C4.D4p: C8⋊1C8 C24.C4 C40.6C4 C56.C4 C88.C4 C104.6C4 ...
(C2×C2p).Q8: C8⋊2C8 C12.53D4 C20.53D4 C28.53D4 C44.53D4 C52.53D4 ...
Matrix representation of C8.C4 ►in GL2(𝔽17) generated by
| 9 | 0 |
| 7 | 2 |
| 2 | 15 |
| 4 | 15 |
G:=sub<GL(2,GF(17))| [9,7,0,2],[2,4,15,15] >;
C8.C4 in GAP, Magma, Sage, TeX
C_8.C_4
% in TeX
G:=Group("C8.C4"); // GroupNames label
G:=SmallGroup(32,15);
// by ID
G=gap.SmallGroup(32,15);
# by ID
G:=PCGroup([5,-2,2,-2,2,-2,40,61,26,302,72,58]);
// Polycyclic
G:=Group<a,b|a^8=1,b^4=a^4,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C8.C4 in TeX
Character table of C8.C4 in TeX