metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C43⋊C3, SmallGroup(129,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C43⋊C3 |
Generators and relations for C43⋊C3
G = < a,b | a43=b3=1, bab-1=a6 >
Character table of C43⋊C3
| class | 1 | 3A | 3B | 43A | 43B | 43C | 43D | 43E | 43F | 43G | 43H | 43I | 43J | 43K | 43L | 43M | 43N | |
| size | 1 | 43 | 43 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | ζ4334+ζ4332+ζ4320 | ζ4341+ζ4331+ζ4314 | ζ4339+ζ4328+ζ4319 | ζ4322+ζ4318+ζ433 | ζ4317+ζ4316+ζ4310 | ζ4338+ζ4335+ζ4313 | ζ4324+ζ4315+ζ434 | ζ4336+ζ436+ζ43 | ζ4329+ζ4312+ζ432 | ζ4333+ζ4327+ζ4326 | ζ4323+ζ4311+ζ439 | ζ4330+ζ438+ζ435 | ζ4340+ζ4325+ζ4321 | ζ4342+ζ4337+ζ437 | complex faithful |
| ρ5 | 3 | 0 | 0 | ζ4323+ζ4311+ζ439 | ζ4329+ζ4312+ζ432 | ζ4324+ζ4315+ζ434 | ζ4340+ζ4325+ζ4321 | ζ4333+ζ4327+ζ4326 | ζ4330+ζ438+ζ435 | ζ4339+ζ4328+ζ4319 | ζ4342+ζ4337+ζ437 | ζ4341+ζ4331+ζ4314 | ζ4317+ζ4316+ζ4310 | ζ4334+ζ4332+ζ4320 | ζ4338+ζ4335+ζ4313 | ζ4322+ζ4318+ζ433 | ζ4336+ζ436+ζ43 | complex faithful |
| ρ6 | 3 | 0 | 0 | ζ4336+ζ436+ζ43 | ζ4330+ζ438+ζ435 | ζ4317+ζ4316+ζ4310 | ζ4341+ζ4331+ζ4314 | ζ4322+ζ4318+ζ433 | ζ4334+ζ4332+ζ4320 | ζ4333+ζ4327+ζ4326 | ζ4339+ζ4328+ζ4319 | ζ4338+ζ4335+ζ4313 | ζ4340+ζ4325+ζ4321 | ζ4342+ζ4337+ζ437 | ζ4323+ζ4311+ζ439 | ζ4329+ζ4312+ζ432 | ζ4324+ζ4315+ζ434 | complex faithful |
| ρ7 | 3 | 0 | 0 | ζ4322+ζ4318+ζ433 | ζ4324+ζ4315+ζ434 | ζ4330+ζ438+ζ435 | ζ4342+ζ4337+ζ437 | ζ4323+ζ4311+ζ439 | ζ4317+ζ4316+ζ4310 | ζ4338+ζ4335+ζ4313 | ζ4341+ζ4331+ζ4314 | ζ4339+ζ4328+ζ4319 | ζ4334+ζ4332+ζ4320 | ζ4340+ζ4325+ζ4321 | ζ4333+ζ4327+ζ4326 | ζ4336+ζ436+ζ43 | ζ4329+ζ4312+ζ432 | complex faithful |
| ρ8 | 3 | 0 | 0 | ζ4342+ζ4337+ζ437 | ζ4338+ζ4335+ζ4313 | ζ4333+ζ4327+ζ4326 | ζ4329+ζ4312+ζ432 | ζ4340+ζ4325+ζ4321 | ζ4323+ζ4311+ζ439 | ζ4317+ζ4316+ζ4310 | ζ4324+ζ4315+ζ434 | ζ4330+ζ438+ζ435 | ζ4322+ζ4318+ζ433 | ζ4336+ζ436+ζ43 | ζ4334+ζ4332+ζ4320 | ζ4341+ζ4331+ζ4314 | ζ4339+ζ4328+ζ4319 | complex faithful |
| ρ9 | 3 | 0 | 0 | ζ4340+ζ4325+ζ4321 | ζ4339+ζ4328+ζ4319 | ζ4338+ζ4335+ζ4313 | ζ4336+ζ436+ζ43 | ζ4334+ζ4332+ζ4320 | ζ4333+ζ4327+ζ4326 | ζ4330+ζ438+ζ435 | ζ4329+ζ4312+ζ432 | ζ4324+ζ4315+ζ434 | ζ4323+ζ4311+ζ439 | ζ4322+ζ4318+ζ433 | ζ4317+ζ4316+ζ4310 | ζ4342+ζ4337+ζ437 | ζ4341+ζ4331+ζ4314 | complex faithful |
| ρ10 | 3 | 0 | 0 | ζ4317+ζ4316+ζ4310 | ζ4342+ζ4337+ζ437 | ζ4341+ζ4331+ζ4314 | ζ4323+ζ4311+ζ439 | ζ4330+ζ438+ζ435 | ζ4339+ζ4328+ζ4319 | ζ4329+ζ4312+ζ432 | ζ4322+ζ4318+ζ433 | ζ4336+ζ436+ζ43 | ζ4338+ζ4335+ζ4313 | ζ4333+ζ4327+ζ4326 | ζ4324+ζ4315+ζ434 | ζ4334+ζ4332+ζ4320 | ζ4340+ζ4325+ζ4321 | complex faithful |
| ρ11 | 3 | 0 | 0 | ζ4338+ζ4335+ζ4313 | ζ4322+ζ4318+ζ433 | ζ4336+ζ436+ζ43 | ζ4317+ζ4316+ζ4310 | ζ4339+ζ4328+ζ4319 | ζ4329+ζ4312+ζ432 | ζ4342+ζ4337+ζ437 | ζ4334+ζ4332+ζ4320 | ζ4340+ζ4325+ζ4321 | ζ4324+ζ4315+ζ434 | ζ4330+ζ438+ζ435 | ζ4341+ζ4331+ζ4314 | ζ4333+ζ4327+ζ4326 | ζ4323+ζ4311+ζ439 | complex faithful |
| ρ12 | 3 | 0 | 0 | ζ4329+ζ4312+ζ432 | ζ4317+ζ4316+ζ4310 | ζ4334+ζ4332+ζ4320 | ζ4339+ζ4328+ζ4319 | ζ4336+ζ436+ζ43 | ζ4340+ζ4325+ζ4321 | ζ4323+ζ4311+ζ439 | ζ4338+ζ4335+ζ4313 | ζ4333+ζ4327+ζ4326 | ζ4342+ζ4337+ζ437 | ζ4341+ζ4331+ζ4314 | ζ4322+ζ4318+ζ433 | ζ4324+ζ4315+ζ434 | ζ4330+ζ438+ζ435 | complex faithful |
| ρ13 | 3 | 0 | 0 | ζ4330+ζ438+ζ435 | ζ4340+ζ4325+ζ4321 | ζ4342+ζ4337+ζ437 | ζ4333+ζ4327+ζ4326 | ζ4324+ζ4315+ζ434 | ζ4341+ζ4331+ζ4314 | ζ4336+ζ436+ζ43 | ζ4323+ζ4311+ζ439 | ζ4322+ζ4318+ζ433 | ζ4339+ζ4328+ζ4319 | ζ4338+ζ4335+ζ4313 | ζ4329+ζ4312+ζ432 | ζ4317+ζ4316+ζ4310 | ζ4334+ζ4332+ζ4320 | complex faithful |
| ρ14 | 3 | 0 | 0 | ζ4339+ζ4328+ζ4319 | ζ4323+ζ4311+ζ439 | ζ4322+ζ4318+ζ433 | ζ4330+ζ438+ζ435 | ζ4341+ζ4331+ζ4314 | ζ4336+ζ436+ζ43 | ζ4340+ζ4325+ζ4321 | ζ4317+ζ4316+ζ4310 | ζ4334+ζ4332+ζ4320 | ζ4329+ζ4312+ζ432 | ζ4324+ζ4315+ζ434 | ζ4342+ζ4337+ζ437 | ζ4338+ζ4335+ζ4313 | ζ4333+ζ4327+ζ4326 | complex faithful |
| ρ15 | 3 | 0 | 0 | ζ4341+ζ4331+ζ4314 | ζ4333+ζ4327+ζ4326 | ζ4323+ζ4311+ζ439 | ζ4324+ζ4315+ζ434 | ζ4342+ζ4337+ζ437 | ζ4322+ζ4318+ζ433 | ζ4334+ζ4332+ζ4320 | ζ4330+ζ438+ζ435 | ζ4317+ζ4316+ζ4310 | ζ4336+ζ436+ζ43 | ζ4329+ζ4312+ζ432 | ζ4340+ζ4325+ζ4321 | ζ4339+ζ4328+ζ4319 | ζ4338+ζ4335+ζ4313 | complex faithful |
| ρ16 | 3 | 0 | 0 | ζ4333+ζ4327+ζ4326 | ζ4336+ζ436+ζ43 | ζ4329+ζ4312+ζ432 | ζ4334+ζ4332+ζ4320 | ζ4338+ζ4335+ζ4313 | ζ4324+ζ4315+ζ434 | ζ4341+ζ4331+ζ4314 | ζ4340+ζ4325+ζ4321 | ζ4342+ζ4337+ζ437 | ζ4330+ζ438+ζ435 | ζ4317+ζ4316+ζ4310 | ζ4339+ζ4328+ζ4319 | ζ4323+ζ4311+ζ439 | ζ4322+ζ4318+ζ433 | complex faithful |
| ρ17 | 3 | 0 | 0 | ζ4324+ζ4315+ζ434 | ζ4334+ζ4332+ζ4320 | ζ4340+ζ4325+ζ4321 | ζ4338+ζ4335+ζ4313 | ζ4329+ζ4312+ζ432 | ζ4342+ζ4337+ζ437 | ζ4322+ζ4318+ζ433 | ζ4333+ζ4327+ζ4326 | ζ4323+ζ4311+ζ439 | ζ4341+ζ4331+ζ4314 | ζ4339+ζ4328+ζ4319 | ζ4336+ζ436+ζ43 | ζ4330+ζ438+ζ435 | ζ4317+ζ4316+ζ4310 | complex 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 37 7)(3 30 13)(4 23 19)(5 16 25)(6 9 31)(8 38 43)(10 24 12)(11 17 18)(14 39 36)(15 32 42)(20 40 29)(21 33 35)(22 26 41)(27 34 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), (2,37,7)(3,30,13)(4,23,19)(5,16,25)(6,9,31)(8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)>;
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), (2,37,7)(3,30,13)(4,23,19)(5,16,25)(6,9,31)(8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28) );
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)], [(2,37,7),(3,30,13),(4,23,19),(5,16,25),(6,9,31),(8,38,43),(10,24,12),(11,17,18),(14,39,36),(15,32,42),(20,40,29),(21,33,35),(22,26,41),(27,34,28)]])
C43⋊C3 is a maximal subgroup of
C43⋊C6
C43⋊C3 is a maximal quotient of C43⋊C9
Matrix representation of C43⋊C3 ►in GL3(𝔽1033) generated by
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 831 | 476 |
| 1 | 0 | 0 |
| 679 | 378 | 614 |
| 92 | 51 | 654 |
G:=sub<GL(3,GF(1033))| [0,0,1,1,0,831,0,1,476],[1,679,92,0,378,51,0,614,654] >;
C43⋊C3 in GAP, Magma, Sage, TeX
C_{43}\rtimes C_3 % in TeX
G:=Group("C43:C3"); // GroupNames label
G:=SmallGroup(129,1);
// by ID
G=gap.SmallGroup(129,1);
# by ID
G:=PCGroup([2,-3,-43,433]);
// Polycyclic
G:=Group<a,b|a^43=b^3=1,b*a*b^-1=a^6>;
// generators/relations
Export
Subgroup lattice of C43⋊C3 in TeX
Character table of C43⋊C3 in TeX