direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×Q16, C8.C6, C24.2C2, C6.16D4, Q8.2C6, C12.19C22, C4.3(C2×C6), C2.5(C3×D4), (C3×Q8).2C2, SmallGroup(48,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q16
G = < a,b,c | a3=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C3×Q16
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | trivial |
| ρ2 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ3 | ζ65 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 6 |
| ρ9 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 6 |
| ρ10 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ32 | ζ6 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ11 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ15 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ16 | 2 | 2 | -1+√-3 | -1-√-3 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ17 | 2 | 2 | -1-√-3 | -1+√-3 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×D4 |
| ρ18 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | complex faithful |
| ρ19 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | complex faithful |
| ρ20 | 2 | -2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | complex faithful |
| ρ21 | 2 | -2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | complex faithful |
►Regular action on 48 points
Generators in S48
(1 34 47)(2 35 48)(3 36 41)(4 37 42)(5 38 43)(6 39 44)(7 40 45)(8 33 46)(9 19 32)(10 20 25)(11 21 26)(12 22 27)(13 23 28)(14 24 29)(15 17 30)(16 18 31)
(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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 45 13 41)(10 44 14 48)(11 43 15 47)(12 42 16 46)(25 39 29 35)(26 38 30 34)(27 37 31 33)(28 36 32 40)
G:=sub<Sym(48)| (1,34,47)(2,35,48)(3,36,41)(4,37,42)(5,38,43)(6,39,44)(7,40,45)(8,33,46)(9,19,32)(10,20,25)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,17,30)(16,18,31), (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), (1,21,5,17)(2,20,6,24)(3,19,7,23)(4,18,8,22)(9,45,13,41)(10,44,14,48)(11,43,15,47)(12,42,16,46)(25,39,29,35)(26,38,30,34)(27,37,31,33)(28,36,32,40)>;
G:=Group( (1,34,47)(2,35,48)(3,36,41)(4,37,42)(5,38,43)(6,39,44)(7,40,45)(8,33,46)(9,19,32)(10,20,25)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,17,30)(16,18,31), (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), (1,21,5,17)(2,20,6,24)(3,19,7,23)(4,18,8,22)(9,45,13,41)(10,44,14,48)(11,43,15,47)(12,42,16,46)(25,39,29,35)(26,38,30,34)(27,37,31,33)(28,36,32,40) );
G=PermutationGroup([[(1,34,47),(2,35,48),(3,36,41),(4,37,42),(5,38,43),(6,39,44),(7,40,45),(8,33,46),(9,19,32),(10,20,25),(11,21,26),(12,22,27),(13,23,28),(14,24,29),(15,17,30),(16,18,31)], [(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)], [(1,21,5,17),(2,20,6,24),(3,19,7,23),(4,18,8,22),(9,45,13,41),(10,44,14,48),(11,43,15,47),(12,42,16,46),(25,39,29,35),(26,38,30,34),(27,37,31,33),(28,36,32,40)]])
C3×Q16 is a maximal subgroup of
C8.6D6 C3⋊Q32 Q16⋊S3 D24⋊C2 Q16.A4 C8.F7 Q8.2F7
C3×Q16 is a maximal quotient of C8.F7 Q8.2F7
Matrix representation of C3×Q16 ►in GL2(𝔽7) generated by
| 4 | 0 |
| 0 | 4 |
| 0 | 6 |
| 1 | 3 |
| 6 | 5 |
| 1 | 1 |
G:=sub<GL(2,GF(7))| [4,0,0,4],[0,1,6,3],[6,1,5,1] >;
C3×Q16 in GAP, Magma, Sage, TeX
C_3\times Q_{16} % in TeX
G:=Group("C3xQ16"); // GroupNames label
G:=SmallGroup(48,27);
// by ID
G=gap.SmallGroup(48,27);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-2,120,141,126,723,368,58]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C3×Q16 in TeX
Character table of C3×Q16 in TeX