Aliases: He3⋊4Dic3, C3⋊(He3⋊C4), (C3×He3)⋊2C4, He3⋊C2.2S3, C3.2(C33⋊C4), C32.2(C32⋊C4), (C3×He3⋊C2).2C2, SmallGroup(324,113)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C3 — C32 — C3×He3 — C3×He3⋊C2 — He3⋊4Dic3 |
C3×He3 — He3⋊4Dic3 |
Generators and relations for He3⋊4Dic3
G = < a,b,c,d,e | a3=b3=c3=d6=1, e2=d3, ab=ba, cac-1=ab-1, dad-1=a-1b, eae-1=a-1bc-1, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece-1=a-1bc, ede-1=d-1 >
Character table of He3⋊4Dic3
class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | |
size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 12 | 12 | 27 | 27 | 9 | 9 | 18 | 18 | 18 | 27 | 27 | 27 | 27 | |
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 | 1 | i | -i | -1 | -1 | -1 | -1 | -1 | i | -i | i | -i | linear of order 4 |
ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | -1 | -i | i | -i | i | linear of order 4 |
ρ5 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
ρ6 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
ρ7 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ3 | ζ32 | ζ32 | ζ3 | complex lifted from He3⋊C4 |
ρ8 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ32 | ζ3 | ζ3 | ζ32 | complex lifted from He3⋊C4 |
ρ9 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | complex lifted from He3⋊C4 |
ρ10 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | complex lifted from He3⋊C4 |
ρ11 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | complex lifted from He3⋊C4 |
ρ12 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | complex lifted from He3⋊C4 |
ρ13 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | complex lifted from He3⋊C4 |
ρ14 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | complex lifted from He3⋊C4 |
ρ15 | 4 | 0 | 4 | 4 | 4 | 4 | 4 | -2 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
ρ16 | 4 | 0 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C4 |
ρ17 | 4 | 0 | 4 | 4 | -2 | -2 | -2 | -1-3√-3/2 | -1+3√-3/2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
ρ18 | 4 | 0 | 4 | 4 | -2 | -2 | -2 | 1 | 1 | 1 | -1+3√-3/2 | -1-3√-3/2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
ρ19 | 4 | 0 | 4 | 4 | -2 | -2 | -2 | 1 | 1 | 1 | -1-3√-3/2 | -1+3√-3/2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
ρ20 | 4 | 0 | 4 | 4 | -2 | -2 | -2 | -1+3√-3/2 | -1-3√-3/2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊C4 |
ρ21 | 6 | 2 | -3+3√-3 | -3-3√-3 | 3+3√-3/2 | 3-3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | complex faithful |
ρ22 | 6 | 2 | -3-3√-3 | -3+3√-3 | 3-3√-3/2 | 3+3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | complex faithful |
ρ23 | 6 | -2 | -3+3√-3 | -3-3√-3 | 3+3√-3/2 | 3-3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
ρ24 | 6 | -2 | -3-3√-3 | -3+3√-3 | 3-3√-3/2 | 3+3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
(1 14 13)(2 18 17)(3 16 15)(4 5 6)(7 9 11)
(1 2 3)(4 6 5)(7 9 11)(8 10 12)(13 17 15)(14 18 16)
(1 15 14)(2 13 18)(3 17 16)(4 11 12)(5 9 10)(6 7 8)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 6)(2 5)(3 4)(7 13 10 16)(8 18 11 15)(9 17 12 14)
G:=sub<Sym(18)| (1,14,13)(2,18,17)(3,16,15)(4,5,6)(7,9,11), (1,2,3)(4,6,5)(7,9,11)(8,10,12)(13,17,15)(14,18,16), (1,15,14)(2,13,18)(3,17,16)(4,11,12)(5,9,10)(6,7,8), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,13,10,16)(8,18,11,15)(9,17,12,14)>;
G:=Group( (1,14,13)(2,18,17)(3,16,15)(4,5,6)(7,9,11), (1,2,3)(4,6,5)(7,9,11)(8,10,12)(13,17,15)(14,18,16), (1,15,14)(2,13,18)(3,17,16)(4,11,12)(5,9,10)(6,7,8), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,6)(2,5)(3,4)(7,13,10,16)(8,18,11,15)(9,17,12,14) );
G=PermutationGroup([[(1,14,13),(2,18,17),(3,16,15),(4,5,6),(7,9,11)], [(1,2,3),(4,6,5),(7,9,11),(8,10,12),(13,17,15),(14,18,16)], [(1,15,14),(2,13,18),(3,17,16),(4,11,12),(5,9,10),(6,7,8)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,6),(2,5),(3,4),(7,13,10,16),(8,18,11,15),(9,17,12,14)]])
G:=TransitiveGroup(18,131);
Matrix representation of He3⋊4Dic3 ►in GL5(𝔽13)
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 3 | 0 |
0 | 0 | 4 | 4 | 7 |
0 | 0 | 12 | 9 | 9 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 3 | 0 | 0 |
0 | 0 | 0 | 3 | 0 |
0 | 0 | 0 | 0 | 3 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 12 | 12 | 8 |
0 | 0 | 3 | 0 | 0 |
0 | 0 | 10 | 1 | 1 |
0 | 12 | 0 | 0 | 0 |
1 | 12 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 12 | 12 | 8 |
0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 11 | 0 | 5 |
0 | 0 | 5 | 5 | 5 |
0 | 0 | 2 | 0 | 2 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,4,12,0,0,3,4,9,0,0,0,7,9],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,1,0,0,0,0,0,12,3,10,0,0,12,0,1,0,0,8,0,1],[0,1,0,0,0,12,12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,8,1],[0,1,0,0,0,1,0,0,0,0,0,0,11,5,2,0,0,0,5,0,0,0,5,5,2] >;
He3⋊4Dic3 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes_4{\rm Dic}_3
% in TeX
G:=Group("He3:4Dic3");
// GroupNames label
G:=SmallGroup(324,113);
// by ID
G=gap.SmallGroup(324,113);
# by ID
G:=PCGroup([6,-2,-2,-3,3,-3,-3,12,362,80,2979,1593,1383,2164]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=1,e^2=d^3,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b,e*a*e^-1=a^-1*b*c^-1,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b*c^-1,e*c*e^-1=a^-1*b*c,e*d*e^-1=d^-1>;
// generators/relations
Export
Subgroup lattice of He3⋊4Dic3 in TeX
Character table of He3⋊4Dic3 in TeX