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## G = C7⋊C3order 21 = 3·7

### The semidirect product of C7 and C3 acting faithfully

Aliases: C7⋊C3, SmallGroup(21,1)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C7⋊C3
G = < a,b | a7=b3=1, bab-1=a4 >

Character table of C7⋊C3

 class 1 3A 3B 7A 7B size 1 7 7 3 3 ρ1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 1 1 linear of order 3 ρ3 1 ζ3 ζ32 1 1 linear of order 3 ρ4 3 0 0 -1-√-7/2 -1+√-7/2 complex faithful ρ5 3 0 0 -1+√-7/2 -1-√-7/2 complex faithful

Permutation representations of C7⋊C3
On 7 points: primitive - transitive group 7T3
Generators in S7
```(1 2 3 4 5 6 7)
(2 3 5)(4 7 6)```

`G:=sub<Sym(7)| (1,2,3,4,5,6,7), (2,3,5)(4,7,6)>;`

`G:=Group( (1,2,3,4,5,6,7), (2,3,5)(4,7,6) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7)], [(2,3,5),(4,7,6)]])`

`G:=TransitiveGroup(7,3);`

Regular action on 21 points - transitive group 21T2
Generators in S21
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 19 10)(2 21 14)(3 16 11)(4 18 8)(5 20 12)(6 15 9)(7 17 13)```

`G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,19,10)(2,21,14)(3,16,11)(4,18,8)(5,20,12)(6,15,9)(7,17,13)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,19,10)(2,21,14)(3,16,11)(4,18,8)(5,20,12)(6,15,9)(7,17,13) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,19,10),(2,21,14),(3,16,11),(4,18,8),(5,20,12),(6,15,9),(7,17,13)]])`

`G:=TransitiveGroup(21,2);`

C7⋊C3 is a maximal subgroup of
F7  C7⋊A4  C49⋊C3  C72⋊C3  C723C3  GL3(𝔽2)  AΓL1(𝔽8)  C91⋊C3  C914C3  C133⋊C3  C1334C3
C7⋊C3 is a maximal quotient of
C7⋊C9  C7⋊A4  C49⋊C3  C72⋊C3  C723C3  AΓL1(𝔽8)  C91⋊C3  C914C3  C133⋊C3  C1334C3

Polynomial with Galois group C7⋊C3 over ℚ
actionf(x)Disc(f)
7T3x7-8x5-2x4+16x3+6x2-6x-226·734

Matrix representation of C7⋊C3 in GL3(𝔽2) generated by

 1 1 0 0 0 1 1 0 0
,
 1 0 0 0 0 1 0 1 1
`G:=sub<GL(3,GF(2))| [1,0,1,1,0,0,0,1,0],[1,0,0,0,0,1,0,1,1] >;`

C7⋊C3 in GAP, Magma, Sage, TeX

`C_7\rtimes C_3`
`% in TeX`

`G:=Group("C7:C3");`
`// GroupNames label`

`G:=SmallGroup(21,1);`
`// by ID`

`G=gap.SmallGroup(21,1);`
`# by ID`

`G:=PCGroup([2,-3,-7,25]);`
`// Polycyclic`

`G:=Group<a,b|a^7=b^3=1,b*a*b^-1=a^4>;`
`// generators/relations`

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