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## G = C31⋊C5order 155 = 5·31

### The semidirect product of C31 and C5 acting faithfully

Aliases: C31⋊C5, SmallGroup(155,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C31⋊C5
 Chief series C1 — C31 — C31⋊C5
 Lower central C31 — C31⋊C5
 Upper central C1

Generators and relations for C31⋊C5
G = < a,b | a31=b5=1, bab-1=a2 >

Character table of C31⋊C5

 class 1 5A 5B 5C 5D 31A 31B 31C 31D 31E 31F size 1 31 31 31 31 5 5 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ53 ζ54 ζ5 ζ52 1 1 1 1 1 1 linear of order 5 ρ3 1 ζ5 ζ53 ζ52 ζ54 1 1 1 1 1 1 linear of order 5 ρ4 1 ζ52 ζ5 ζ54 ζ53 1 1 1 1 1 1 linear of order 5 ρ5 1 ζ54 ζ52 ζ53 ζ5 1 1 1 1 1 1 linear of order 5 ρ6 5 0 0 0 0 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 complex faithful ρ7 5 0 0 0 0 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3120+ζ3118+ζ3110+ζ319+ζ315 complex faithful ρ8 5 0 0 0 0 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 complex faithful ρ9 5 0 0 0 0 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3124+ζ3117+ζ3112+ζ316+ζ313 complex faithful ρ10 5 0 0 0 0 ζ3116+ζ318+ζ314+ζ312+ζ31 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 complex faithful ρ11 5 0 0 0 0 ζ3124+ζ3117+ζ3112+ζ316+ζ313 ζ3126+ζ3122+ζ3121+ζ3113+ζ3111 ζ3120+ζ3118+ζ3110+ζ319+ζ315 ζ3130+ζ3129+ζ3127+ζ3123+ζ3115 ζ3128+ζ3125+ζ3119+ζ3114+ζ317 ζ3116+ζ318+ζ314+ζ312+ζ31 complex faithful

Permutation representations of C31⋊C5
On 31 points: primitive - transitive group 31T4
Generators in S31
```(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)
(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)```

`G:=sub<Sym(31)| (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), (2,17,9,5,3)(4,18,25,13,7)(6,19,10,21,11)(8,20,26,29,15)(12,22,27,14,23)(16,24,28,30,31)>;`

`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), (2,17,9,5,3)(4,18,25,13,7)(6,19,10,21,11)(8,20,26,29,15)(12,22,27,14,23)(16,24,28,30,31) );`

`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)], [(2,17,9,5,3),(4,18,25,13,7),(6,19,10,21,11),(8,20,26,29,15),(12,22,27,14,23),(16,24,28,30,31)]])`

`G:=TransitiveGroup(31,4);`

C31⋊C5 is a maximal subgroup of   C31⋊C10  C31⋊C15

Matrix representation of C31⋊C5 in GL5(𝔽2)

 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0
,
 1 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1

`G:=sub<GL(5,GF(2))| [0,0,0,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,0,0,0,1,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,1] >;`

C31⋊C5 in GAP, Magma, Sage, TeX

`C_{31}\rtimes C_5`
`% in TeX`

`G:=Group("C31:C5");`
`// GroupNames label`

`G:=SmallGroup(155,1);`
`// by ID`

`G=gap.SmallGroup(155,1);`
`# by ID`

`G:=PCGroup([2,-5,-31,321]);`
`// Polycyclic`

`G:=Group<a,b|a^31=b^5=1,b*a*b^-1=a^2>;`
`// generators/relations`

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