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## G = C31⋊C6order 186 = 2·3·31

### The semidirect product of C31 and C6 acting faithfully

Aliases: C31⋊C6, D31⋊C3, C31⋊C3⋊C2, SmallGroup(186,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C31 — C31⋊C6
 Chief series C1 — C31 — C31⋊C3 — C31⋊C6
 Lower central C31 — C31⋊C6
 Upper central C1

Generators and relations for C31⋊C6
G = < a,b | a31=b6=1, bab-1=a6 >

Character table of C31⋊C6

 class 1 2 3A 3B 6A 6B 31A 31B 31C 31D 31E size 1 31 31 31 31 31 6 6 6 6 6 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 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 linear of order 6 ρ4 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 linear of order 3 ρ5 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 linear of order 6 ρ6 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 linear of order 3 ρ7 6 0 0 0 0 0 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 orthogonal faithful ρ8 6 0 0 0 0 0 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 orthogonal faithful ρ9 6 0 0 0 0 0 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 orthogonal faithful ρ10 6 0 0 0 0 0 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 orthogonal faithful ρ11 6 0 0 0 0 0 ζ3127+ζ3124+ζ3120+ζ3111+ζ317+ζ314 ζ3123+ζ3122+ζ3117+ζ3114+ζ319+ζ318 ζ3129+ζ3121+ζ3119+ζ3112+ζ3110+ζ312 ζ3128+ζ3118+ζ3116+ζ3115+ζ3113+ζ313 ζ3130+ζ3126+ζ3125+ζ316+ζ315+ζ31 orthogonal faithful

Permutation representations of C31⋊C6
On 31 points: primitive - transitive group 31T5
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 27 26 31 6 7)(3 22 20 30 11 13)(4 17 14 29 16 19)(5 12 8 28 21 25)(9 23 15 24 10 18)```

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

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

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

`G:=TransitiveGroup(31,5);`

C31⋊C6 is a maximal quotient of   C31⋊C12

Matrix representation of C31⋊C6 in GL6(𝔽373)

 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 372 57 291 217 291 57
,
 1 0 0 0 0 0 372 57 291 217 291 57 195 252 72 78 328 276 220 4 365 365 4 220 276 328 78 72 252 195 57 291 217 291 57 372

`G:=sub<GL(6,GF(373))| [0,0,0,0,0,372,1,0,0,0,0,57,0,1,0,0,0,291,0,0,1,0,0,217,0,0,0,1,0,291,0,0,0,0,1,57],[1,372,195,220,276,57,0,57,252,4,328,291,0,291,72,365,78,217,0,217,78,365,72,291,0,291,328,4,252,57,0,57,276,220,195,372] >;`

C31⋊C6 in GAP, Magma, Sage, TeX

`C_{31}\rtimes C_6`
`% in TeX`

`G:=Group("C31:C6");`
`// GroupNames label`

`G:=SmallGroup(186,1);`
`// by ID`

`G=gap.SmallGroup(186,1);`
`# by ID`

`G:=PCGroup([3,-2,-3,-31,1622,680]);`
`// Polycyclic`

`G:=Group<a,b|a^31=b^6=1,b*a*b^-1=a^6>;`
`// generators/relations`

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