metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊2D16, D8⋊1S3, C6.8D8, C8.4D6, D24⋊3C2, C12.3D4, C24.2C22, C3⋊C16⋊1C2, (C3×D8)⋊1C2, C2.4(D4⋊S3), C4.1(C3⋊D4), SmallGroup(96,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊D16
G = < a,b,c | a3=b16=c2=1, bab-1=cac=a-1, cbc=b-1 >
Character table of C3⋊D16
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 8A | 8B | 12 | 16A | 16B | 16C | 16D | 24A | 24B | |
| size | 1 | 1 | 8 | 24 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 1 | 1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ8 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | √2 | -√2 | 0 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | √2 | -√2 | orthogonal lifted from D16 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -√2 | √2 | 0 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | -ζ165+ζ163 | -√2 | √2 | orthogonal lifted from D16 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | √2 | -√2 | 0 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -ζ1615+ζ169 | √2 | -√2 | orthogonal lifted from D16 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | -√2 | √2 | 0 | -ζ1615+ζ169 | -ζ165+ζ163 | ζ1615-ζ169 | ζ165-ζ163 | -√2 | √2 | orthogonal lifted from D16 |
| ρ14 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -√-3 | √-3 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ15 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | √-3 | -√-3 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 4 | 4 | 0 | 0 | -2 | -4 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | orthogonal faithful, Schur index 2 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | orthogonal faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 34 22)(2 23 35)(3 36 24)(4 25 37)(5 38 26)(6 27 39)(7 40 28)(8 29 41)(9 42 30)(10 31 43)(11 44 32)(12 17 45)(13 46 18)(14 19 47)(15 48 20)(16 21 33)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)
G:=sub<Sym(48)| (1,34,22)(2,23,35)(3,36,24)(4,25,37)(5,38,26)(6,27,39)(7,40,28)(8,29,41)(9,42,30)(10,31,43)(11,44,32)(12,17,45)(13,46,18)(14,19,47)(15,48,20)(16,21,33), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)>;
G:=Group( (1,34,22)(2,23,35)(3,36,24)(4,25,37)(5,38,26)(6,27,39)(7,40,28)(8,29,41)(9,42,30)(10,31,43)(11,44,32)(12,17,45)(13,46,18)(14,19,47)(15,48,20)(16,21,33), (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), (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40) );
G=PermutationGroup([[(1,34,22),(2,23,35),(3,36,24),(4,25,37),(5,38,26),(6,27,39),(7,40,28),(8,29,41),(9,42,30),(10,31,43),(11,44,32),(12,17,45),(13,46,18),(14,19,47),(15,48,20),(16,21,33)], [(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)], [(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40)]])
C3⋊D16 is a maximal subgroup of
S3×D16 D8⋊D6 D48⋊C2 D6.2D8 D8.D6 Q16⋊D6 Q16.D6 C9⋊D16 C32⋊2D16 C3⋊D48 C32⋊7D16 C15⋊D16 C3⋊D80 C15⋊7D16
C3⋊D16 is a maximal quotient of
C6.6D16 C6.D16 C3⋊D32 D16.S3 C3⋊SD64 C3⋊Q64 D8⋊1Dic3 C9⋊D16 C32⋊2D16 C3⋊D48 C32⋊7D16 C15⋊D16 C3⋊D80 C15⋊7D16
Matrix representation of C3⋊D16 ►in GL4(𝔽7) generated by
| 5 | 6 | 3 | 6 |
| 1 | 3 | 3 | 3 |
| 4 | 3 | 2 | 6 |
| 0 | 0 | 0 | 2 |
| 3 | 5 | 2 | 3 |
| 3 | 1 | 1 | 5 |
| 5 | 3 | 2 | 0 |
| 2 | 0 | 3 | 1 |
| 2 | 2 | 3 | 2 |
| 6 | 5 | 3 | 5 |
| 1 | 1 | 3 | 2 |
| 5 | 2 | 0 | 4 |
G:=sub<GL(4,GF(7))| [5,1,4,0,6,3,3,0,3,3,2,0,6,3,6,2],[3,3,5,2,5,1,3,0,2,1,2,3,3,5,0,1],[2,6,1,5,2,5,1,2,3,3,3,0,2,5,2,4] >;
C3⋊D16 in GAP, Magma, Sage, TeX
C_3\rtimes D_{16} % in TeX
G:=Group("C3:D16"); // GroupNames label
G:=SmallGroup(96,33);
// by ID
G=gap.SmallGroup(96,33);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,218,116,122,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C3⋊D16 in TeX
Character table of C3⋊D16 in TeX