non-abelian, soluble, monomial
Aliases: 2+ 1+4.1C6, (C22×C4)⋊2A4, C23.7D4⋊C3, C23.2(C2×A4), C23⋊A4.1C2, C2.5(C24⋊C6), SmallGroup(192,202)
Series: Derived ►Chief ►Lower central ►Upper central
| 2+ 1+4 — 2+ 1+4.C6 |
Generators and relations for 2+ 1+4.C6
G = < a,b,c,d,e | a4=b2=d2=1, c2=e6=a2, bab=ece-1=a-1, ac=ca, ad=da, eae-1=acd, bc=cb, bd=db, ebe-1=a-1c, dcd=a2c, ede-1=a-1bd >
Character table of 2+ 1+4.C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 6 | 12 | 16 | 16 | 4 | 4 | 12 | 24 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 3 | 3 | 3 | -1 | 0 | 0 | -3 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ8 | 3 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 4 | -4 | 0 | 0 | 1 | 1 | 2i | -2i | 0 | 0 | -1 | -1 | i | -i | i | -i | complex faithful |
| ρ10 | 4 | -4 | 0 | 0 | 1 | 1 | -2i | 2i | 0 | 0 | -1 | -1 | -i | i | -i | i | complex faithful |
| ρ11 | 4 | -4 | 0 | 0 | ζ32 | ζ3 | -2i | 2i | 0 | 0 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | complex faithful |
| ρ12 | 4 | -4 | 0 | 0 | ζ3 | ζ32 | 2i | -2i | 0 | 0 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | complex faithful |
| ρ13 | 4 | -4 | 0 | 0 | ζ3 | ζ32 | -2i | 2i | 0 | 0 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | complex faithful |
| ρ14 | 4 | -4 | 0 | 0 | ζ32 | ζ3 | 2i | -2i | 0 | 0 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | complex faithful |
| ρ15 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C6 |
| ρ16 | 6 | 6 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C6 |
►On 16 points - transitive group 16T426
Generators in S16
(1 11 3 5)(2 14 4 8)(6 16 12 10)(7 15 13 9)
(1 5)(2 8)(3 11)(4 14)(6 10)(7 15)(9 13)(12 16)
(1 15 3 9)(2 6 4 12)(5 7 11 13)(8 10 14 16)
(1 3)(2 4)(5 11)(8 14)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,11,3,5)(2,14,4,8)(6,16,12,10)(7,15,13,9), (1,5)(2,8)(3,11)(4,14)(6,10)(7,15)(9,13)(12,16), (1,15,3,9)(2,6,4,12)(5,7,11,13)(8,10,14,16), (1,3)(2,4)(5,11)(8,14), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,11,3,5)(2,14,4,8)(6,16,12,10)(7,15,13,9), (1,5)(2,8)(3,11)(4,14)(6,10)(7,15)(9,13)(12,16), (1,15,3,9)(2,6,4,12)(5,7,11,13)(8,10,14,16), (1,3)(2,4)(5,11)(8,14), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,11,3,5),(2,14,4,8),(6,16,12,10),(7,15,13,9)], [(1,5),(2,8),(3,11),(4,14),(6,10),(7,15),(9,13),(12,16)], [(1,15,3,9),(2,6,4,12),(5,7,11,13),(8,10,14,16)], [(1,3),(2,4),(5,11),(8,14)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,426);
►On 16 points - transitive group 16T428
Generators in S16
(1 14 3 8)(2 9 4 15)(5 13 11 7)(6 16 12 10)
(2 13)(4 7)(5 9)(6 12)(8 14)(11 15)
(1 10 3 16)(2 9 4 15)(5 7 11 13)(6 8 12 14)
(1 16)(2 13)(3 10)(4 7)(5 15)(6 8)(9 11)(12 14)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,14,3,8)(2,9,4,15)(5,13,11,7)(6,16,12,10), (2,13)(4,7)(5,9)(6,12)(8,14)(11,15), (1,10,3,16)(2,9,4,15)(5,7,11,13)(6,8,12,14), (1,16)(2,13)(3,10)(4,7)(5,15)(6,8)(9,11)(12,14), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16)>;
G:=Group( (1,14,3,8)(2,9,4,15)(5,13,11,7)(6,16,12,10), (2,13)(4,7)(5,9)(6,12)(8,14)(11,15), (1,10,3,16)(2,9,4,15)(5,7,11,13)(6,8,12,14), (1,16)(2,13)(3,10)(4,7)(5,15)(6,8)(9,11)(12,14), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,14,3,8),(2,9,4,15),(5,13,11,7),(6,16,12,10)], [(2,13),(4,7),(5,9),(6,12),(8,14),(11,15)], [(1,10,3,16),(2,9,4,15),(5,7,11,13),(6,8,12,14)], [(1,16),(2,13),(3,10),(4,7),(5,15),(6,8),(9,11),(12,14)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,428);
Matrix representation of 2+ 1+4.C6 ►in GL4(𝔽5) generated by
| 0 | 2 | 1 | 3 |
| 2 | 1 | 3 | 0 |
| 1 | 3 | 0 | 3 |
| 3 | 0 | 3 | 4 |
| 1 | 0 | 0 | 0 |
| 3 | 4 | 2 | 0 |
| 0 | 0 | 1 | 0 |
| 3 | 0 | 3 | 4 |
| 0 | 0 | 4 | 0 |
| 2 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 3 | 4 | 2 | 0 |
| 4 | 0 | 0 | 0 |
| 2 | 0 | 2 | 1 |
| 0 | 0 | 1 | 0 |
| 2 | 1 | 3 | 0 |
| 3 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
G:=sub<GL(4,GF(5))| [0,2,1,3,2,1,3,0,1,3,0,3,3,0,3,4],[1,3,0,3,0,4,0,0,0,2,1,3,0,0,0,4],[0,2,1,3,0,0,0,4,4,2,0,2,0,1,0,0],[4,2,0,2,0,0,0,1,0,2,1,3,0,1,0,0],[3,0,3,4,4,0,0,0,0,4,0,0,0,0,4,0] >;
2+ 1+4.C6 in GAP, Magma, Sage, TeX
2_+^{1+4}.C_6 % in TeX
G:=Group("ES+(2,2).C6"); // GroupNames label
G:=SmallGroup(192,202);
// by ID
G=gap.SmallGroup(192,202);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,632,135,1683,262,851,375,3540,1027]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=d^2=1,c^2=e^6=a^2,b*a*b=e*c*e^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*c,d*c*d=a^2*c,e*d*e^-1=a^-1*b*d>;
// generators/relations
Export
Subgroup lattice of 2+ 1+4.C6 in TeX
Character table of 2+ 1+4.C6 in TeX