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## G = C61⋊C5order 305 = 5·61

### The semidirect product of C61 and C5 acting faithfully

Aliases: C61⋊C5, SmallGroup(305,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C61 — C61⋊C5
 Chief series C1 — C61 — C61⋊C5
 Lower central C61 — C61⋊C5
 Upper central C1

Generators and relations for C61⋊C5
G = < a,b | a61=b5=1, bab-1=a34 >

Character table of C61⋊C5

 class 1 5A 5B 5C 5D 61A 61B 61C 61D 61E 61F 61G 61H 61I 61J 61K 61L size 1 61 61 61 61 5 5 5 5 5 5 5 5 5 5 5 5 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ5 ζ53 ζ52 ζ54 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ3 1 ζ54 ζ52 ζ53 ζ5 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ4 1 ζ52 ζ5 ζ54 ζ53 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ5 1 ζ53 ζ54 ζ5 ζ52 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 5 ρ6 5 0 0 0 0 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 complex faithful ρ7 5 0 0 0 0 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 complex faithful ρ8 5 0 0 0 0 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 complex faithful ρ9 5 0 0 0 0 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 complex faithful ρ10 5 0 0 0 0 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 complex faithful ρ11 5 0 0 0 0 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 complex faithful ρ12 5 0 0 0 0 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6158+ζ6134+ζ6120+ζ619+ζ61 complex faithful ρ13 5 0 0 0 0 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 complex faithful ρ14 5 0 0 0 0 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 complex faithful ρ15 5 0 0 0 0 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 complex faithful ρ16 5 0 0 0 0 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 ζ6155+ζ6140+ζ6118+ζ617+ζ612 complex faithful ρ17 5 0 0 0 0 ζ6155+ζ6140+ζ6118+ζ617+ζ612 ζ6148+ζ6146+ζ6145+ζ6139+ζ615 ζ6160+ζ6152+ζ6141+ζ6127+ζ613 ζ6149+ζ6136+ζ6119+ζ6114+ζ614 ζ6135+ζ6131+ζ6129+ζ6117+ζ6110 ζ6159+ζ6154+ζ6143+ζ6121+ζ616 ζ6158+ζ6134+ζ6120+ζ619+ζ61 ζ6157+ζ6147+ζ6142+ζ6125+ζ6112 ζ6138+ζ6137+ζ6128+ζ6111+ζ618 ζ6156+ζ6122+ζ6116+ζ6115+ζ6113 ζ6153+ζ6150+ζ6133+ζ6124+ζ6123 ζ6151+ζ6144+ζ6132+ζ6130+ζ6126 complex faithful

Smallest permutation representation of C61⋊C5
On 61 points: primitive
Generators in S61
```(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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61)
(2 10 21 59 35)(3 19 41 56 8)(4 28 61 53 42)(5 37 20 50 15)(6 46 40 47 49)(7 55 60 44 22)(9 12 39 38 29)(11 30 18 32 36)(13 48 58 26 43)(14 57 17 23 16)(24 25 34 54 51)(27 52 33 45 31)```

`G:=sub<Sym(61)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61), (2,10,21,59,35)(3,19,41,56,8)(4,28,61,53,42)(5,37,20,50,15)(6,46,40,47,49)(7,55,60,44,22)(9,12,39,38,29)(11,30,18,32,36)(13,48,58,26,43)(14,57,17,23,16)(24,25,34,54,51)(27,52,33,45,31)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61), (2,10,21,59,35)(3,19,41,56,8)(4,28,61,53,42)(5,37,20,50,15)(6,46,40,47,49)(7,55,60,44,22)(9,12,39,38,29)(11,30,18,32,36)(13,48,58,26,43)(14,57,17,23,16)(24,25,34,54,51)(27,52,33,45,31) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)], [(2,10,21,59,35),(3,19,41,56,8),(4,28,61,53,42),(5,37,20,50,15),(6,46,40,47,49),(7,55,60,44,22),(9,12,39,38,29),(11,30,18,32,36),(13,48,58,26,43),(14,57,17,23,16),(24,25,34,54,51),(27,52,33,45,31)]])`

Matrix representation of C61⋊C5 in GL5(𝔽1831)

 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1110 1287 1215 848
,
 1 0 0 0 0 1340 1602 1813 393 151 736 1182 752 91 511 975 607 1232 675 601 114 1164 133 1323 632

`G:=sub<GL(5,GF(1831))| [0,0,0,0,1,1,0,0,0,1110,0,1,0,0,1287,0,0,1,0,1215,0,0,0,1,848],[1,1340,736,975,114,0,1602,1182,607,1164,0,1813,752,1232,133,0,393,91,675,1323,0,151,511,601,632] >;`

C61⋊C5 in GAP, Magma, Sage, TeX

`C_{61}\rtimes C_5`
`% in TeX`

`G:=Group("C61:C5");`
`// GroupNames label`

`G:=SmallGroup(305,1);`
`// by ID`

`G=gap.SmallGroup(305,1);`
`# by ID`

`G:=PCGroup([2,-5,-61,181]);`
`// Polycyclic`

`G:=Group<a,b|a^61=b^5=1,b*a*b^-1=a^34>;`
`// generators/relations`

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