metacyclic, supersoluble, monomial, Z-group
Aliases: C43⋊C6, D43⋊C3, C43⋊C3⋊C2, SmallGroup(258,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C43⋊C6 |
Generators and relations for C43⋊C6
G = < a,b | a43=b6=1, bab-1=a7 >
Character table of C43⋊C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 43A | 43B | 43C | 43D | 43E | 43F | 43G | |
| size | 1 | 43 | 43 | 43 | 43 | 43 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | orthogonal faithful |
| ρ8 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | orthogonal faithful |
| ρ9 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | orthogonal faithful |
| ρ10 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | orthogonal faithful |
| ρ12 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | orthogonal faithful |
| ρ13 | 6 | 0 | 0 | 0 | 0 | 0 | ζ4339+ζ4328+ζ4324+ζ4319+ζ4315+ζ434 | ζ4341+ζ4331+ζ4329+ζ4314+ζ4312+ζ432 | ζ4333+ζ4327+ζ4326+ζ4317+ζ4316+ζ4310 | ζ4334+ζ4332+ζ4323+ζ4320+ζ4311+ζ439 | ζ4338+ζ4335+ζ4330+ζ4313+ζ438+ζ435 | ζ4340+ζ4325+ζ4322+ζ4321+ζ4318+ζ433 | ζ4342+ζ4337+ζ4336+ζ437+ζ436+ζ43 | orthogonal faithful |
►On 43 points: primitive
Generators in S43
(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)
(2 38 37 43 7 8)(3 32 30 42 13 15)(4 26 23 41 19 22)(5 20 16 40 25 29)(6 14 9 39 31 36)(10 33 24 35 12 21)(11 27 17 34 18 28)
G:=sub<Sym(43)| (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), (2,38,37,43,7,8)(3,32,30,42,13,15)(4,26,23,41,19,22)(5,20,16,40,25,29)(6,14,9,39,31,36)(10,33,24,35,12,21)(11,27,17,34,18,28)>;
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), (2,38,37,43,7,8)(3,32,30,42,13,15)(4,26,23,41,19,22)(5,20,16,40,25,29)(6,14,9,39,31,36)(10,33,24,35,12,21)(11,27,17,34,18,28) );
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)], [(2,38,37,43,7,8),(3,32,30,42,13,15),(4,26,23,41,19,22),(5,20,16,40,25,29),(6,14,9,39,31,36),(10,33,24,35,12,21),(11,27,17,34,18,28)]])
Matrix representation of C43⋊C6 ►in GL6(𝔽1033)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1032 | 678 | 272 | 868 | 272 | 678 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 355 | 1031 | 187 | 999 | 377 | 271 |
| 1029 | 922 | 1011 | 516 | 520 | 1007 |
| 196 | 174 | 312 | 872 | 801 | 193 |
| 837 | 857 | 397 | 555 | 941 | 838 |
| 4 | 361 | 465 | 143 | 956 | 276 |
G:=sub<GL(6,GF(1033))| [0,0,0,0,0,1032,1,0,0,0,0,678,0,1,0,0,0,272,0,0,1,0,0,868,0,0,0,1,0,272,0,0,0,0,1,678],[1,355,1029,196,837,4,0,1031,922,174,857,361,0,187,1011,312,397,465,0,999,516,872,555,143,0,377,520,801,941,956,0,271,1007,193,838,276] >;
C43⋊C6 in GAP, Magma, Sage, TeX
C_{43}\rtimes C_6 % in TeX
G:=Group("C43:C6"); // GroupNames label
G:=SmallGroup(258,1);
// by ID
G=gap.SmallGroup(258,1);
# by ID
G:=PCGroup([3,-2,-3,-43,2270,977]);
// Polycyclic
G:=Group<a,b|a^43=b^6=1,b*a*b^-1=a^7>;
// generators/relations
Export
Subgroup lattice of C43⋊C6 in TeX
Character table of C43⋊C6 in TeX