metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic12, C8.S3, C3⋊1Q16, C6.3D4, C24.1C2, C4.10D6, C2.5D12, Dic6.1C2, C12.10C22, SmallGroup(48,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic12
G = < a,b | a24=1, b2=a12, bab-1=a-1 >
Character table of Dic12
class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
size | 1 | 1 | 2 | 2 | 12 | 12 | 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 | trivial |
ρ2 | 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 | linear of order 2 |
ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ6 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ7 | 2 | 2 | -1 | 2 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ8 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
ρ9 | 2 | 2 | -1 | -2 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
ρ10 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | -√2 | √2 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
ρ11 | 2 | -2 | 2 | 0 | 0 | 0 | -2 | √2 | -√2 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
ρ12 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √2 | -√2 | √3 | -√3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | symplectic faithful, Schur index 2 |
ρ13 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√2 | √2 | √3 | -√3 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | symplectic faithful, Schur index 2 |
ρ14 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | -√2 | √2 | -√3 | √3 | ζ87ζ32+ζ85ζ32+ζ85 | ζ83ζ32+ζ83+ζ8ζ32 | ζ83ζ3+ζ8ζ3+ζ8 | ζ87ζ3+ζ87+ζ85ζ3 | symplectic faithful, Schur index 2 |
ρ15 | 2 | -2 | -1 | 0 | 0 | 0 | 1 | √2 | -√2 | -√3 | √3 | ζ87ζ3+ζ87+ζ85ζ3 | ζ83ζ3+ζ8ζ3+ζ8 | ζ83ζ32+ζ83+ζ8ζ32 | ζ87ζ32+ζ85ζ32+ζ85 | symplectic faithful, Schur index 2 |
(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 47 13 35)(2 46 14 34)(3 45 15 33)(4 44 16 32)(5 43 17 31)(6 42 18 30)(7 41 19 29)(8 40 20 28)(9 39 21 27)(10 38 22 26)(11 37 23 25)(12 36 24 48)
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), (1,47,13,35)(2,46,14,34)(3,45,15,33)(4,44,16,32)(5,43,17,31)(6,42,18,30)(7,41,19,29)(8,40,20,28)(9,39,21,27)(10,38,22,26)(11,37,23,25)(12,36,24,48)>;
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), (1,47,13,35)(2,46,14,34)(3,45,15,33)(4,44,16,32)(5,43,17,31)(6,42,18,30)(7,41,19,29)(8,40,20,28)(9,39,21,27)(10,38,22,26)(11,37,23,25)(12,36,24,48) );
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)], [(1,47,13,35),(2,46,14,34),(3,45,15,33),(4,44,16,32),(5,43,17,31),(6,42,18,30),(7,41,19,29),(8,40,20,28),(9,39,21,27),(10,38,22,26),(11,37,23,25),(12,36,24,48)]])
Dic12 is a maximal subgroup of
C48⋊C2 Dic24 D8.S3 C3⋊Q32 C4○D24 C8.D6 D8⋊3S3 D4.D6 S3×Q16 Dic36 C32⋊3Q16 C32⋊5Q16 A4⋊Q16 C8.S4 C5⋊Dic12 Dic60 C7⋊Dic12 Dic84 He3⋊Q16 C33⋊3Q16
Dic12 is a maximal quotient of
C2.Dic12 C24⋊1C4 Dic36 C32⋊3Q16 C32⋊5Q16 A4⋊Q16 C5⋊Dic12 Dic60 C7⋊Dic12 Dic84 C33⋊3Q16
Matrix representation of Dic12 ►in GL2(𝔽23) generated by
9 | 17 |
17 | 22 |
0 | 22 |
1 | 0 |
G:=sub<GL(2,GF(23))| [9,17,17,22],[0,1,22,0] >;
Dic12 in GAP, Magma, Sage, TeX
{\rm Dic}_{12}
% in TeX
G:=Group("Dic12");
// GroupNames label
G:=SmallGroup(48,8);
// by ID
G=gap.SmallGroup(48,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,40,61,66,182,42,804]);
// Polycyclic
G:=Group<a,b|a^24=1,b^2=a^12,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic12 in TeX
Character table of Dic12 in TeX