direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×Q8, C6.12C23, C12.20C22, C4.4(C2×C6), (C2×C4).3C6, (C2×C12).9C2, C22.4(C2×C6), C2.2(C22×C6), (C2×C6).15C22, SmallGroup(48,46)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Q8
G = < a,b,c | a6=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C6×Q8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | -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 | 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 | -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 | 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 | -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 | 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 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ12 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ32 | ζ3 | linear of order 6 |
| ρ13 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | ζ6 | ζ3 | ζ32 | linear of order 6 |
| ρ14 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | ζ65 | ζ32 | ζ3 | linear of order 6 |
| ρ15 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | 1 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ16 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ17 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ3 | ζ32 | linear of order 6 |
| ρ18 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ19 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | 1 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ20 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ21 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ6 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ22 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | -1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | ζ3 | ζ32 | ζ3 | linear of order 6 |
| ρ23 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | -1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | ζ65 | ζ6 | ζ32 | ζ3 | ζ32 | linear of order 6 |
| ρ24 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | ζ32 | ζ6 | ζ65 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ26 | 2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ27 | 2 | -2 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | -1-√-3 | 1+√-3 | -1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ28 | 2 | 2 | -2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 1-√-3 | -1+√-3 | 1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ29 | 2 | 2 | -2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 1+√-3 | -1-√-3 | 1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
| ρ30 | 2 | -2 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | -1+√-3 | 1-√-3 | -1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×Q8 |
►Regular action on 48 points
Generators in S48
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 29 16 19)(2 30 17 20)(3 25 18 21)(4 26 13 22)(5 27 14 23)(6 28 15 24)(7 37 48 33)(8 38 43 34)(9 39 44 35)(10 40 45 36)(11 41 46 31)(12 42 47 32)
(1 41 16 31)(2 42 17 32)(3 37 18 33)(4 38 13 34)(5 39 14 35)(6 40 15 36)(7 21 48 25)(8 22 43 26)(9 23 44 27)(10 24 45 28)(11 19 46 29)(12 20 47 30)
G:=sub<Sym(48)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,29,16,19)(2,30,17,20)(3,25,18,21)(4,26,13,22)(5,27,14,23)(6,28,15,24)(7,37,48,33)(8,38,43,34)(9,39,44,35)(10,40,45,36)(11,41,46,31)(12,42,47,32), (1,41,16,31)(2,42,17,32)(3,37,18,33)(4,38,13,34)(5,39,14,35)(6,40,15,36)(7,21,48,25)(8,22,43,26)(9,23,44,27)(10,24,45,28)(11,19,46,29)(12,20,47,30)>;
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,29,16,19)(2,30,17,20)(3,25,18,21)(4,26,13,22)(5,27,14,23)(6,28,15,24)(7,37,48,33)(8,38,43,34)(9,39,44,35)(10,40,45,36)(11,41,46,31)(12,42,47,32), (1,41,16,31)(2,42,17,32)(3,37,18,33)(4,38,13,34)(5,39,14,35)(6,40,15,36)(7,21,48,25)(8,22,43,26)(9,23,44,27)(10,24,45,28)(11,19,46,29)(12,20,47,30) );
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,29,16,19),(2,30,17,20),(3,25,18,21),(4,26,13,22),(5,27,14,23),(6,28,15,24),(7,37,48,33),(8,38,43,34),(9,39,44,35),(10,40,45,36),(11,41,46,31),(12,42,47,32)], [(1,41,16,31),(2,42,17,32),(3,37,18,33),(4,38,13,34),(5,39,14,35),(6,40,15,36),(7,21,48,25),(8,22,43,26),(9,23,44,27),(10,24,45,28),(11,19,46,29),(12,20,47,30)]])
C6×Q8 is a maximal subgroup of
Q8⋊2Dic3 C12.10D4 Q8.11D6 Dic3⋊Q8 D6⋊3Q8 C12.23D4 Q8.15D6
Matrix representation of C6×Q8 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 3 |
| 0 | 0 | 3 | 4 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[12,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4] >;
C6×Q8 in GAP, Magma, Sage, TeX
C_6\times Q_8
% in TeX
G:=Group("C6xQ8"); // GroupNames label
G:=SmallGroup(48,46);
// by ID
G=gap.SmallGroup(48,46);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-2,120,261,126]);
// Polycyclic
G:=Group<a,b,c|a^6=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C6×Q8 in TeX
Character table of C6×Q8 in TeX