p-group, metacyclic, nilpotent (class 4), monomial
Aliases: C8.Q8, C16⋊1C4, C4.9SD16, M5(2).1C2, C22.5SD16, C4.6(C4⋊C4), C8.19(C2×C4), (C2×C4).13D4, C4.Q8.1C2, C2.3(C4.Q8), C8.C4.2C2, (C2×C8).12C22, 2-Sylow(AGammaL(1,81)), SmallGroup(64,46)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.Q8
G = < a,b,c | a8=1, b4=a2, c2=a-1b2, bab-1=a5, cac-1=a3, cbc-1=a4b3 >
Character table of C8.Q8
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | i | -i | 1 | 1 | -1 | -i | i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -i | i | 1 | 1 | -1 | -i | i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | i | -i | 1 | 1 | -1 | i | -i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -i | i | 1 | 1 | -1 | i | -i | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ13 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ14 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T136
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)
G:=sub<Sym(16)| (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15)>;
G:=Group( (1,3,5,7,9,11,13,15)(2,12,6,16,10,4,14,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,4,10,12)(3,7)(5,13)(6,16,14,8)(11,15) );
G=PermutationGroup([[(1,3,5,7,9,11,13,15),(2,12,6,16,10,4,14,8)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,4,10,12),(3,7),(5,13),(6,16,14,8),(11,15)]])
G:=TransitiveGroup(16,136);
C8.Q8 is a maximal subgroup of
M5(2)⋊3C4 Q32⋊C4 D16⋊C4 M5(2).C22 C23.10SD16 C80⋊4C4 C80⋊5C4
C4p.SD16: D8⋊3Q8 D8.2Q8 C8.Dic6 C24.6Q8 C24.Q8 C8.Dic10 C40.6Q8 C40.Q8 ...
C8.Q8 is a maximal quotient of
C16⋊1C8
C4p.(C4⋊C4): C8.11C42 C8.Dic6 C24.6Q8 C24.Q8 C8.Dic10 C40.6Q8 C40.Q8 C80⋊4C4 ...
Matrix representation of C8.Q8 ►in GL4(𝔽3) generated by
| 2 | 0 | 1 | 0 |
| 0 | 2 | 0 | 1 |
| 2 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 2 | 0 | 1 |
| 2 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(3))| [2,0,2,0,0,2,0,1,1,0,2,0,0,1,0,0],[0,0,0,2,0,0,2,0,0,2,0,1,1,0,1,0],[1,0,0,0,0,0,0,1,0,0,2,0,0,2,0,0] >;
C8.Q8 in GAP, Magma, Sage, TeX
C_8.Q_8
% in TeX
G:=Group("C8.Q8"); // GroupNames label
G:=SmallGroup(64,46);
// by ID
G=gap.SmallGroup(64,46);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,362,86,489,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=a^2,c^2=a^-1*b^2,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=a^4*b^3>;
// generators/relations
Export
Subgroup lattice of C8.Q8 in TeX
Character table of C8.Q8 in TeX