metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3⋊C16, C6.C8, C8.2S3, C24.3C2, C12.2C4, C4.2Dic3, C2.(C3⋊C8), SmallGroup(48,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C3⋊C16 |
Generators and relations for C3⋊C16
G = < a,b | a3=b16=1, bab-1=a-1 >
Character table of C3⋊C16
| class | 1 | 2 | 3 | 4A | 4B | 6 | 8A | 8B | 8C | 8D | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | i | -i | -i | -i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | -i | i | i | i | -i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | -i | i | -i | i | linear of order 8 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | i | -i | i | -i | linear of order 8 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | -i | i | -i | i | linear of order 8 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | i | -i | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | i | -i | i | -i | linear of order 8 |
| ρ9 | 1 | -1 | 1 | -i | i | -1 | ζ166 | ζ1610 | ζ162 | ζ1614 | -i | i | ζ167 | ζ16 | ζ1613 | ζ165 | ζ1611 | ζ163 | ζ1615 | ζ169 | ζ1610 | ζ166 | ζ162 | ζ1614 | linear of order 16 |
| ρ10 | 1 | -1 | 1 | i | -i | -1 | ζ162 | ζ1614 | ζ166 | ζ1610 | i | -i | ζ165 | ζ163 | ζ167 | ζ1615 | ζ16 | ζ169 | ζ1613 | ζ1611 | ζ1614 | ζ162 | ζ166 | ζ1610 | linear of order 16 |
| ρ11 | 1 | -1 | 1 | i | -i | -1 | ζ1610 | ζ166 | ζ1614 | ζ162 | i | -i | ζ16 | ζ167 | ζ1611 | ζ163 | ζ1613 | ζ165 | ζ169 | ζ1615 | ζ166 | ζ1610 | ζ1614 | ζ162 | linear of order 16 |
| ρ12 | 1 | -1 | 1 | -i | i | -1 | ζ1614 | ζ162 | ζ1610 | ζ166 | -i | i | ζ163 | ζ165 | ζ16 | ζ169 | ζ167 | ζ1615 | ζ1611 | ζ1613 | ζ162 | ζ1614 | ζ1610 | ζ166 | linear of order 16 |
| ρ13 | 1 | -1 | 1 | i | -i | -1 | ζ162 | ζ1614 | ζ166 | ζ1610 | i | -i | ζ1613 | ζ1611 | ζ1615 | ζ167 | ζ169 | ζ16 | ζ165 | ζ163 | ζ1614 | ζ162 | ζ166 | ζ1610 | linear of order 16 |
| ρ14 | 1 | -1 | 1 | i | -i | -1 | ζ1610 | ζ166 | ζ1614 | ζ162 | i | -i | ζ169 | ζ1615 | ζ163 | ζ1611 | ζ165 | ζ1613 | ζ16 | ζ167 | ζ166 | ζ1610 | ζ1614 | ζ162 | linear of order 16 |
| ρ15 | 1 | -1 | 1 | -i | i | -1 | ζ1614 | ζ162 | ζ1610 | ζ166 | -i | i | ζ1611 | ζ1613 | ζ169 | ζ16 | ζ1615 | ζ167 | ζ163 | ζ165 | ζ162 | ζ1614 | ζ1610 | ζ166 | linear of order 16 |
| ρ16 | 1 | -1 | 1 | -i | i | -1 | ζ166 | ζ1610 | ζ162 | ζ1614 | -i | i | ζ1615 | ζ169 | ζ165 | ζ1613 | ζ163 | ζ1611 | ζ167 | ζ16 | ζ1610 | ζ166 | ζ162 | ζ1614 | linear of order 16 |
| ρ17 | 2 | 2 | -1 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | -1 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ19 | 2 | 2 | -1 | -2 | -2 | -1 | 2i | -2i | -2i | 2i | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from C3⋊C8 |
| ρ20 | 2 | 2 | -1 | -2 | -2 | -1 | -2i | 2i | 2i | -2i | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from C3⋊C8 |
| ρ21 | 2 | -2 | -1 | 2i | -2i | 1 | 2ζ85 | 2ζ83 | 2ζ87 | 2ζ8 | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ87 | ζ8 | ζ83 | ζ85 | complex faithful, Schur index 2 |
| ρ22 | 2 | -2 | -1 | 2i | -2i | 1 | 2ζ8 | 2ζ87 | 2ζ83 | 2ζ85 | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83 | ζ85 | ζ87 | ζ8 | complex faithful, Schur index 2 |
| ρ23 | 2 | -2 | -1 | -2i | 2i | 1 | 2ζ87 | 2ζ8 | 2ζ85 | 2ζ83 | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ85 | ζ83 | ζ8 | ζ87 | complex faithful, Schur index 2 |
| ρ24 | 2 | -2 | -1 | -2i | 2i | 1 | 2ζ83 | 2ζ85 | 2ζ8 | 2ζ87 | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ8 | ζ87 | ζ85 | ζ83 | complex faithful, Schur index 2 |
►Regular action on 48 points
Generators in S48
(1 41 24)(2 25 42)(3 43 26)(4 27 44)(5 45 28)(6 29 46)(7 47 30)(8 31 48)(9 33 32)(10 17 34)(11 35 18)(12 19 36)(13 37 20)(14 21 38)(15 39 22)(16 23 40)
(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)
G:=sub<Sym(48)| (1,41,24)(2,25,42)(3,43,26)(4,27,44)(5,45,28)(6,29,46)(7,47,30)(8,31,48)(9,33,32)(10,17,34)(11,35,18)(12,19,36)(13,37,20)(14,21,38)(15,39,22)(16,23,40), (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)>;
G:=Group( (1,41,24)(2,25,42)(3,43,26)(4,27,44)(5,45,28)(6,29,46)(7,47,30)(8,31,48)(9,33,32)(10,17,34)(11,35,18)(12,19,36)(13,37,20)(14,21,38)(15,39,22)(16,23,40), (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) );
G=PermutationGroup([[(1,41,24),(2,25,42),(3,43,26),(4,27,44),(5,45,28),(6,29,46),(7,47,30),(8,31,48),(9,33,32),(10,17,34),(11,35,18),(12,19,36),(13,37,20),(14,21,38),(15,39,22),(16,23,40)], [(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)]])
C3⋊C16 is a maximal subgroup of
S3×C16 D6.C8 C12.C8 C3⋊D16 D8.S3 C8.6D6 C3⋊Q32 C9⋊C16 C24.S3 A4⋊C16 C8.7S4 C15⋊3C16 C15⋊C16 C21⋊C16 He3⋊C16 C33⋊4C16 C6.F9
C3⋊C16 is a maximal quotient of
C3⋊C32 C9⋊C16 C24.S3 A4⋊C16 C15⋊3C16 C15⋊C16 C21⋊C16 C33⋊4C16 C6.F9
Matrix representation of C3⋊C16 ►in GL2(𝔽17) generated by
| 16 | 6 |
| 14 | 0 |
| 6 | 15 |
| 0 | 11 |
G:=sub<GL(2,GF(17))| [16,14,6,0],[6,0,15,11] >;
C3⋊C16 in GAP, Magma, Sage, TeX
C_3\rtimes C_{16} % in TeX
G:=Group("C3:C16"); // GroupNames label
G:=SmallGroup(48,1);
// by ID
G=gap.SmallGroup(48,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,10,26,42,804]);
// Polycyclic
G:=Group<a,b|a^3=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C3⋊C16 in TeX
Character table of C3⋊C16 in TeX