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## G = C37⋊C9order 333 = 32·37

### The semidirect product of C37 and C9 acting faithfully

Aliases: C37⋊C9, C37⋊C3.C3, SmallGroup(333,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C37 — C37⋊C9
 Chief series C1 — C37 — C37⋊C3 — C37⋊C9
 Lower central C37 — C37⋊C9
 Upper central C1

Generators and relations for C37⋊C9
G = < a,b | a37=b9=1, bab-1=a7 >

Character table of C37⋊C9

 class 1 3A 3B 9A 9B 9C 9D 9E 9F 37A 37B 37C 37D size 1 37 37 37 37 37 37 37 37 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 linear of order 3 ρ3 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 linear of order 3 ρ4 1 ζ32 ζ3 ζ98 ζ9 ζ94 ζ97 ζ92 ζ95 1 1 1 1 linear of order 9 ρ5 1 ζ32 ζ3 ζ92 ζ97 ζ9 ζ94 ζ95 ζ98 1 1 1 1 linear of order 9 ρ6 1 ζ3 ζ32 ζ9 ζ98 ζ95 ζ92 ζ97 ζ94 1 1 1 1 linear of order 9 ρ7 1 ζ3 ζ32 ζ94 ζ95 ζ92 ζ98 ζ9 ζ97 1 1 1 1 linear of order 9 ρ8 1 ζ3 ζ32 ζ97 ζ92 ζ98 ζ95 ζ94 ζ9 1 1 1 1 linear of order 9 ρ9 1 ζ32 ζ3 ζ95 ζ94 ζ97 ζ9 ζ98 ζ92 1 1 1 1 linear of order 9 ρ10 9 0 0 0 0 0 0 0 0 ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 complex faithful ρ11 9 0 0 0 0 0 0 0 0 ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 complex faithful ρ12 9 0 0 0 0 0 0 0 0 ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 complex faithful ρ13 9 0 0 0 0 0 0 0 0 ζ3736+ζ3730+ζ3728+ζ3727+ζ3725+ζ3721+ζ3711+ζ374+ζ373 ζ3734+ζ3733+ζ3726+ζ3716+ζ3712+ζ3710+ζ379+ζ377+ζ37 ζ3732+ζ3731+ζ3729+ζ3724+ζ3720+ζ3718+ζ3715+ζ3714+ζ372 ζ3735+ζ3723+ζ3722+ζ3719+ζ3717+ζ3713+ζ378+ζ376+ζ375 complex faithful

Smallest permutation representation of C37⋊C9
On 37 points: primitive
Generators in S37
```(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)
(2 17 35 27 10 34 11 13 8)(3 33 32 16 19 30 21 25 15)(4 12 29 5 28 26 31 37 22)(6 7 23 20 9 18 14 24 36)```

`G:=sub<Sym(37)| (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), (2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22)(6,7,23,20,9,18,14,24,36)>;`

`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), (2,17,35,27,10,34,11,13,8)(3,33,32,16,19,30,21,25,15)(4,12,29,5,28,26,31,37,22)(6,7,23,20,9,18,14,24,36) );`

`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)], [(2,17,35,27,10,34,11,13,8),(3,33,32,16,19,30,21,25,15),(4,12,29,5,28,26,31,37,22),(6,7,23,20,9,18,14,24,36)]])`

Matrix representation of C37⋊C9 in GL9(𝔽1999)

 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 544 1415 698 705 712 467 1417 621
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 497 1963 368 754 1203 1495 1052 581 463 81 881 540 1146 1583 546 1519 371 495 252 1744 1917 1895 1547 1842 1878 1585 702 668 1446 1196 999 1054 757 612 1993 1455 0 0 0 0 0 1 0 0 0 124 741 1053 1639 797 1818 1816 1383 1460 1171 1092 1262 90 1350 86 1090 139 795

`G:=sub<GL(9,GF(1999))| [0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,544,0,1,0,0,0,0,0,0,1415,0,0,1,0,0,0,0,0,698,0,0,0,1,0,0,0,0,705,0,0,0,0,1,0,0,0,712,0,0,0,0,0,1,0,0,467,0,0,0,0,0,0,1,0,1417,0,0,0,0,0,0,0,1,621],[1,0,497,81,252,668,0,124,1171,0,0,1963,881,1744,1446,0,741,1092,0,0,368,540,1917,1196,0,1053,1262,0,0,754,1146,1895,999,0,1639,90,0,0,1203,1583,1547,1054,0,797,1350,0,0,1495,546,1842,757,1,1818,86,0,0,1052,1519,1878,612,0,1816,1090,0,1,581,371,1585,1993,0,1383,139,0,0,463,495,702,1455,0,1460,795] >;`

C37⋊C9 in GAP, Magma, Sage, TeX

`C_{37}\rtimes C_9`
`% in TeX`

`G:=Group("C37:C9");`
`// GroupNames label`

`G:=SmallGroup(333,3);`
`// by ID`

`G=gap.SmallGroup(333,3);`
`# by ID`

`G:=PCGroup([3,-3,-3,-37,9,1298,707]);`
`// Polycyclic`

`G:=Group<a,b|a^37=b^9=1,b*a*b^-1=a^7>;`
`// generators/relations`

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