metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊2Q16, C4.4D6, C6.10D4, Q8.2S3, C12.4C22, Dic6.2C2, C3⋊C8.C2, (C3×Q8).1C2, C2.7(C3⋊D4), SmallGroup(48,18)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊Q16
G = < a,b,c | a3=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Character table of C3⋊Q16
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 12A | 12B | 12C | |
| size | 1 | 1 | 2 | 2 | 4 | 12 | 2 | 6 | 6 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | -1 | 2 | -2 | 0 | -1 | 0 | 0 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ9 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ10 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | -√-3 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ11 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | √-3 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ12 | 4 | -4 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►Regular action on 48 points
(1 46 31)(2 32 47)(3 48 25)(4 26 41)(5 42 27)(6 28 43)(7 44 29)(8 30 45)(9 19 35)(10 36 20)(11 21 37)(12 38 22)(13 23 39)(14 40 24)(15 17 33)(16 34 18)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 31 13 27)(10 30 14 26)(11 29 15 25)(12 28 16 32)(33 48 37 44)(34 47 38 43)(35 46 39 42)(36 45 40 41)
G:=sub<Sym(48)| (1,46,31)(2,32,47)(3,48,25)(4,26,41)(5,42,27)(6,28,43)(7,44,29)(8,30,45)(9,19,35)(10,36,20)(11,21,37)(12,38,22)(13,23,39)(14,40,24)(15,17,33)(16,34,18), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,31,13,27)(10,30,14,26)(11,29,15,25)(12,28,16,32)(33,48,37,44)(34,47,38,43)(35,46,39,42)(36,45,40,41)>;
G:=Group( (1,46,31)(2,32,47)(3,48,25)(4,26,41)(5,42,27)(6,28,43)(7,44,29)(8,30,45)(9,19,35)(10,36,20)(11,21,37)(12,38,22)(13,23,39)(14,40,24)(15,17,33)(16,34,18), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,31,13,27)(10,30,14,26)(11,29,15,25)(12,28,16,32)(33,48,37,44)(34,47,38,43)(35,46,39,42)(36,45,40,41) );
G=PermutationGroup([[(1,46,31),(2,32,47),(3,48,25),(4,26,41),(5,42,27),(6,28,43),(7,44,29),(8,30,45),(9,19,35),(10,36,20),(11,21,37),(12,38,22),(13,23,39),(14,40,24),(15,17,33),(16,34,18)], [(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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,31,13,27),(10,30,14,26),(11,29,15,25),(12,28,16,32),(33,48,37,44),(34,47,38,43),(35,46,39,42),(36,45,40,41)]])
C3⋊Q16 is a maximal subgroup of
D4.D6 Q8.7D6 S3×Q16 Q16⋊S3 Q8.11D6 Q8.13D6 Q8.14D6 C9⋊Q16 Q8.D9 C32⋊2Q16 C32⋊3Q16 C32⋊7Q16 C6.5S4 A4⋊2Q16 Q8.4S4 C15⋊Q16 C3⋊Dic20 C15⋊7Q16 C21⋊Q16 C3⋊Dic28 C21⋊7Q16 C33⋊Q16
C3⋊Q16 is a maximal quotient of
C6.Q16 C6.SD16 Q8⋊2Dic3 C9⋊Q16 C32⋊2Q16 C32⋊3Q16 C32⋊7Q16 A4⋊2Q16 C15⋊Q16 C3⋊Dic20 C15⋊7Q16 C21⋊Q16 C3⋊Dic28 C21⋊7Q16 C33⋊Q16
Matrix representation of C3⋊Q16 ►in GL4(𝔽5) generated by
| 1 | 0 | 2 | 3 |
| 4 | 2 | 4 | 3 |
| 1 | 4 | 2 | 0 |
| 0 | 4 | 4 | 3 |
| 2 | 2 | 3 | 3 |
| 2 | 4 | 0 | 2 |
| 0 | 3 | 3 | 0 |
| 1 | 4 | 1 | 1 |
| 3 | 2 | 4 | 0 |
| 1 | 2 | 1 | 3 |
| 2 | 0 | 4 | 1 |
| 1 | 1 | 0 | 1 |
G:=sub<GL(4,GF(5))| [1,4,1,0,0,2,4,4,2,4,2,4,3,3,0,3],[2,2,0,1,2,4,3,4,3,0,3,1,3,2,0,1],[3,1,2,1,2,2,0,1,4,1,4,0,0,3,1,1] >;
C3⋊Q16 in GAP, Magma, Sage, TeX
C_3\rtimes Q_{16} % in TeX
G:=Group("C3:Q16"); // GroupNames label
G:=SmallGroup(48,18);
// by ID
G=gap.SmallGroup(48,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,40,61,46,182,97,42,804]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=1,c^2=b^4,b*a*b^-1=a^-1,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