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## G = C43⋊C7order 301 = 7·43

### The semidirect product of C43 and C7 acting faithfully

Aliases: C43⋊C7, SmallGroup(301,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C43 — C43⋊C7
 Chief series C1 — C43 — C43⋊C7
 Lower central C43 — C43⋊C7
 Upper central C1

Generators and relations for C43⋊C7
G = < a,b | a43=b7=1, bab-1=a21 >

Character table of C43⋊C7

 class 1 7A 7B 7C 7D 7E 7F 43A 43B 43C 43D 43E 43F size 1 43 43 43 43 43 43 7 7 7 7 7 7 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 1 1 1 1 1 1 linear of order 7 ρ3 1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 1 1 1 1 1 1 linear of order 7 ρ4 1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 1 1 1 1 1 1 linear of order 7 ρ5 1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 1 1 1 1 1 1 linear of order 7 ρ6 1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 1 1 1 1 1 1 linear of order 7 ρ7 1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 1 1 1 1 1 1 linear of order 7 ρ8 7 0 0 0 0 0 0 ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 complex faithful ρ9 7 0 0 0 0 0 0 ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 complex faithful ρ10 7 0 0 0 0 0 0 ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 complex faithful ρ11 7 0 0 0 0 0 0 ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 complex faithful ρ12 7 0 0 0 0 0 0 ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 complex faithful ρ13 7 0 0 0 0 0 0 ζ4340+ζ4338+ζ4331+ζ4324+ζ4323+ζ4310+ζ436 ζ4337+ζ4333+ζ4320+ζ4319+ζ4312+ζ435+ζ433 ζ4334+ζ4330+ζ4329+ζ4328+ζ4326+ζ4318+ζ437 ζ4336+ζ4325+ζ4317+ζ4315+ζ4314+ζ4313+ζ439 ζ4342+ζ4339+ζ4332+ζ4327+ζ4322+ζ438+ζ432 ζ4341+ζ4335+ζ4321+ζ4316+ζ4311+ζ434+ζ43 complex faithful

Smallest permutation representation of C43⋊C7
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 42 5 36 17 12 22)(3 40 9 28 33 23 43)(4 38 13 20 6 34 21)(7 32 25 39 11 24 41)(8 30 29 31 27 35 19)(10 26 37 15 16 14 18)```

`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), (2,42,5,36,17,12,22)(3,40,9,28,33,23,43)(4,38,13,20,6,34,21)(7,32,25,39,11,24,41)(8,30,29,31,27,35,19)(10,26,37,15,16,14,18)>;`

`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), (2,42,5,36,17,12,22)(3,40,9,28,33,23,43)(4,38,13,20,6,34,21)(7,32,25,39,11,24,41)(8,30,29,31,27,35,19)(10,26,37,15,16,14,18) );`

`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)], [(2,42,5,36,17,12,22),(3,40,9,28,33,23,43),(4,38,13,20,6,34,21),(7,32,25,39,11,24,41),(8,30,29,31,27,35,19),(10,26,37,15,16,14,18)]])`

Matrix representation of C43⋊C7 in GL7(𝔽3011)

 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 195 1916 1073 194 1917 1072
,
 1 0 0 0 0 0 0 2771 3002 2479 372 273 2796 1647 2816 1095 1938 2817 1094 1939 1 1626 1636 1957 1668 2193 2992 2771 2988 2194 1282 2792 2390 1282 2816 721 2926 339 2904 2335 57 1626 657 185 372 830 208 373 2988

`G:=sub<GL(7,GF(3011))| [0,0,0,0,0,0,1,1,0,0,0,0,0,195,0,1,0,0,0,0,1916,0,0,1,0,0,0,1073,0,0,0,1,0,0,194,0,0,0,0,1,0,1917,0,0,0,0,0,1,1072],[1,2771,2816,1626,2988,721,657,0,3002,1095,1636,2194,2926,185,0,2479,1938,1957,1282,339,372,0,372,2817,1668,2792,2904,830,0,273,1094,2193,2390,2335,208,0,2796,1939,2992,1282,57,373,0,1647,1,2771,2816,1626,2988] >;`

C43⋊C7 in GAP, Magma, Sage, TeX

`C_{43}\rtimes C_7`
`% in TeX`

`G:=Group("C43:C7");`
`// GroupNames label`

`G:=SmallGroup(301,1);`
`// by ID`

`G=gap.SmallGroup(301,1);`
`# by ID`

`G:=PCGroup([2,-7,-43,1149]);`
`// Polycyclic`

`G:=Group<a,b|a^43=b^7=1,b*a*b^-1=a^21>;`
`// generators/relations`

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