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## G = C29⋊C4order 116 = 22·29

### The semidirect product of C29 and C4 acting faithfully

Aliases: C29⋊C4, D29.C2, SmallGroup(116,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — C29⋊C4
 Chief series C1 — C29 — D29 — C29⋊C4
 Lower central C29 — C29⋊C4
 Upper central C1

Generators and relations for C29⋊C4
G = < a,b | a29=b4=1, bab-1=a17 >

Character table of C29⋊C4

 class 1 2 4A 4B 29A 29B 29C 29D 29E 29F 29G size 1 29 29 29 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 i -i 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 -i i 1 1 1 1 1 1 1 linear of order 4 ρ5 4 0 0 0 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 orthogonal faithful ρ6 4 0 0 0 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2921+ζ2920+ζ299+ζ298 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 orthogonal faithful ρ7 4 0 0 0 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 orthogonal faithful ρ8 4 0 0 0 ζ2918+ζ2916+ζ2913+ζ2911 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2923+ζ2915+ζ2914+ζ296 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 orthogonal faithful ρ9 4 0 0 0 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2927+ζ2924+ζ295+ζ292 ζ2925+ζ2919+ζ2910+ζ294 ζ2923+ζ2915+ζ2914+ζ296 orthogonal faithful ρ10 4 0 0 0 ζ2923+ζ2915+ζ2914+ζ296 ζ2921+ζ2920+ζ299+ζ298 ζ2928+ζ2917+ζ2912+ζ29 ζ2927+ζ2924+ζ295+ζ292 ζ2918+ζ2916+ζ2913+ζ2911 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 orthogonal faithful ρ11 4 0 0 0 ζ2927+ζ2924+ζ295+ζ292 ζ2926+ζ2922+ζ297+ζ293 ζ2925+ζ2919+ζ2910+ζ294 ζ2921+ζ2920+ζ299+ζ298 ζ2923+ζ2915+ζ2914+ζ296 ζ2928+ζ2917+ζ2912+ζ29 ζ2918+ζ2916+ζ2913+ζ2911 orthogonal faithful

Permutation representations of C29⋊C4
On 29 points: primitive - transitive group 29T3
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 13 29 18)(3 25 28 6)(4 8 27 23)(5 20 26 11)(7 15 24 16)(9 10 22 21)(12 17 19 14)```

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

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

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

`G:=TransitiveGroup(29,3);`

C29⋊C4 is a maximal subgroup of   C87⋊C4
C29⋊C4 is a maximal quotient of   C29⋊C8  C87⋊C4

Matrix representation of C29⋊C4 in GL4(𝔽233) generated by

 232 1 0 0 232 0 1 0 232 0 0 1 216 212 21 16
,
 61 25 47 222 217 212 21 16 57 166 56 137 14 209 189 137
`G:=sub<GL(4,GF(233))| [232,232,232,216,1,0,0,212,0,1,0,21,0,0,1,16],[61,217,57,14,25,212,166,209,47,21,56,189,222,16,137,137] >;`

C29⋊C4 in GAP, Magma, Sage, TeX

`C_{29}\rtimes C_4`
`% in TeX`

`G:=Group("C29:C4");`
`// GroupNames label`

`G:=SmallGroup(116,3);`
`// by ID`

`G=gap.SmallGroup(116,3);`
`# by ID`

`G:=PCGroup([3,-2,-2,-29,6,434,509]);`
`// Polycyclic`

`G:=Group<a,b|a^29=b^4=1,b*a*b^-1=a^17>;`
`// generators/relations`

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