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## G = C43⋊C3order 129 = 3·43

### The semidirect product of C43 and C3 acting faithfully

Aliases: C43⋊C3, SmallGroup(129,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C43 — C43⋊C3
 Chief series C1 — C43 — C43⋊C3
 Lower central C43 — C43⋊C3
 Upper central C1

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

Smallest permutation representation of C43⋊C3
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`

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