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## G = C7⋊C9order 63 = 32·7

### The semidirect product of C7 and C9 acting via C9/C3=C3

Aliases: C7⋊C9, C21.C3, C3.(C7⋊C3), SmallGroup(63,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C7⋊C9
 Chief series C1 — C7 — C21 — C7⋊C9
 Lower central C7 — C7⋊C9
 Upper central C1 — C3

Generators and relations for C7⋊C9
G = < a,b | a7=b9=1, bab-1=a4 >

Character table of C7⋊C9

 class 1 3A 3B 7A 7B 9A 9B 9C 9D 9E 9F 21A 21B 21C 21D size 1 1 1 3 3 7 7 7 7 7 7 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 linear of order 3 ρ3 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 linear of order 3 ρ4 1 ζ3 ζ32 1 1 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 ζ32 ζ32 ζ3 ζ3 linear of order 9 ρ5 1 ζ3 ζ32 1 1 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 ζ32 ζ32 ζ3 ζ3 linear of order 9 ρ6 1 ζ32 ζ3 1 1 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 ζ3 ζ3 ζ32 ζ32 linear of order 9 ρ7 1 ζ3 ζ32 1 1 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 ζ32 ζ32 ζ3 ζ3 linear of order 9 ρ8 1 ζ32 ζ3 1 1 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 ζ3 ζ3 ζ32 ζ32 linear of order 9 ρ9 1 ζ32 ζ3 1 1 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 ζ3 ζ3 ζ32 ζ32 linear of order 9 ρ10 3 3 3 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ11 3 3 3 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ12 3 -3-3√-3/2 -3+3√-3/2 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 complex faithful, Schur index 3 ρ13 3 -3+3√-3/2 -3-3√-3/2 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 complex faithful, Schur index 3 ρ14 3 -3-3√-3/2 -3+3√-3/2 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 complex faithful, Schur index 3 ρ15 3 -3+3√-3/2 -3-3√-3/2 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 complex faithful, Schur index 3

Smallest permutation representation of C7⋊C9
Regular action on 63 points
Generators in S63
```(1 56 16 27 47 29 38)(2 48 57 30 17 39 19)(3 18 49 40 58 20 31)(4 59 10 21 50 32 41)(5 51 60 33 11 42 22)(6 12 52 43 61 23 34)(7 62 13 24 53 35 44)(8 54 63 36 14 45 25)(9 15 46 37 55 26 28)
(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 62 63)```

`G:=sub<Sym(63)| (1,56,16,27,47,29,38)(2,48,57,30,17,39,19)(3,18,49,40,58,20,31)(4,59,10,21,50,32,41)(5,51,60,33,11,42,22)(6,12,52,43,61,23,34)(7,62,13,24,53,35,44)(8,54,63,36,14,45,25)(9,15,46,37,55,26,28), (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)(55,56,57,58,59,60,61,62,63)>;`

`G:=Group( (1,56,16,27,47,29,38)(2,48,57,30,17,39,19)(3,18,49,40,58,20,31)(4,59,10,21,50,32,41)(5,51,60,33,11,42,22)(6,12,52,43,61,23,34)(7,62,13,24,53,35,44)(8,54,63,36,14,45,25)(9,15,46,37,55,26,28), (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)(55,56,57,58,59,60,61,62,63) );`

`G=PermutationGroup([(1,56,16,27,47,29,38),(2,48,57,30,17,39,19),(3,18,49,40,58,20,31),(4,59,10,21,50,32,41),(5,51,60,33,11,42,22),(6,12,52,43,61,23,34),(7,62,13,24,53,35,44),(8,54,63,36,14,45,25),(9,15,46,37,55,26,28)], [(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),(55,56,57,58,59,60,61,62,63)])`

C7⋊C9 is a maximal subgroup of
C7⋊C18  C9×C7⋊C3  C63⋊C3  C633C3  C21.C32  C21.A4  C49⋊C9  C72⋊C9  C723C9
C7⋊C9 is a maximal quotient of
C7⋊C27  C21.A4  C49⋊C9  C72⋊C9  C723C9

Matrix representation of C7⋊C9 in GL3(𝔽127) generated by

 0 1 0 0 0 1 1 23 22
,
 49 2 53 111 53 85 53 125 25
`G:=sub<GL(3,GF(127))| [0,0,1,1,0,23,0,1,22],[49,111,53,2,53,125,53,85,25] >;`

C7⋊C9 in GAP, Magma, Sage, TeX

`C_7\rtimes C_9`
`% in TeX`

`G:=Group("C7:C9");`
`// GroupNames label`

`G:=SmallGroup(63,1);`
`// by ID`

`G=gap.SmallGroup(63,1);`
`# by ID`

`G:=PCGroup([3,-3,-3,-7,9,164]);`
`// Polycyclic`

`G:=Group<a,b|a^7=b^9=1,b*a*b^-1=a^4>;`
`// generators/relations`

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