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## G = C29⋊C7order 203 = 7·29

### The semidirect product of C29 and C7 acting faithfully

Aliases: C29⋊C7, SmallGroup(203,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — C29⋊C7
 Chief series C1 — C29 — C29⋊C7
 Lower central C29 — C29⋊C7
 Upper central C1

Generators and relations for C29⋊C7
G = < a,b | a29=b7=1, bab-1=a20 >

Character table of C29⋊C7

 class 1 7A 7B 7C 7D 7E 7F 29A 29B 29C 29D size 1 29 29 29 29 29 29 7 7 7 7 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 1 1 1 1 linear of order 7 ρ3 1 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 1 1 1 1 linear of order 7 ρ4 1 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 1 1 1 1 linear of order 7 ρ5 1 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 1 1 1 1 linear of order 7 ρ6 1 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 1 1 1 1 linear of order 7 ρ7 1 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 1 1 1 1 linear of order 7 ρ8 7 0 0 0 0 0 0 ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 complex faithful ρ9 7 0 0 0 0 0 0 ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 complex faithful ρ10 7 0 0 0 0 0 0 ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 complex faithful ρ11 7 0 0 0 0 0 0 ζ2921+ζ2919+ζ2917+ζ2914+ζ2911+ζ293+ζ292 ζ2928+ζ2922+ζ2913+ζ299+ζ296+ζ295+ζ294 ζ2927+ζ2926+ζ2918+ζ2915+ζ2912+ζ2910+ζ298 ζ2925+ζ2924+ζ2923+ζ2920+ζ2916+ζ297+ζ29 complex faithful

Permutation representations of C29⋊C7
On 29 points: primitive - transitive group 29T4
Generators in S29
```(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)
(2 17 25 8 26 24 21)(3 4 20 15 22 18 12)(5 7 10 29 14 6 23)(9 13 19 28 27 11 16)```

`G:=sub<Sym(29)| (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), (2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16)>;`

`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), (2,17,25,8,26,24,21)(3,4,20,15,22,18,12)(5,7,10,29,14,6,23)(9,13,19,28,27,11,16) );`

`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)], [(2,17,25,8,26,24,21),(3,4,20,15,22,18,12),(5,7,10,29,14,6,23),(9,13,19,28,27,11,16)]])`

`G:=TransitiveGroup(29,4);`

C29⋊C7 is a maximal subgroup of   C29⋊C14

Matrix representation of C29⋊C7 in GL7(𝔽2437)

 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 1689 2174 481 1905 749 262
,
 1 0 0 0 0 0 0 2356 1874 867 1348 1842 706 148 1234 2283 2360 2299 715 1652 1764 0 0 1 0 0 0 0 490 1615 1608 1243 1461 2056 2248 790 598 1991 1199 1652 1615 932 0 0 0 0 1 0 0

`G:=sub<GL(7,GF(2437))| [0,0,0,0,0,0,1,1,0,0,0,0,0,1689,0,1,0,0,0,0,2174,0,0,1,0,0,0,481,0,0,0,1,0,0,1905,0,0,0,0,1,0,749,0,0,0,0,0,1,262],[1,2356,1234,0,490,790,0,0,1874,2283,0,1615,598,0,0,867,2360,1,1608,1991,0,0,1348,2299,0,1243,1199,0,0,1842,715,0,1461,1652,1,0,706,1652,0,2056,1615,0,0,148,1764,0,2248,932,0] >;`

C29⋊C7 in GAP, Magma, Sage, TeX

`C_{29}\rtimes C_7`
`% in TeX`

`G:=Group("C29:C7");`
`// GroupNames label`

`G:=SmallGroup(203,1);`
`// by ID`

`G=gap.SmallGroup(203,1);`
`# by ID`

`G:=PCGroup([2,-7,-29,449]);`
`// Polycyclic`

`G:=Group<a,b|a^29=b^7=1,b*a*b^-1=a^20>;`
`// generators/relations`

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