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## G = C7⋊C32order 441 = 32·72

### Direct product of C7⋊C3 and C7⋊C3

Aliases: C7⋊C32, C72⋊C32, C723C3⋊C3, C72⋊C3⋊C3, (C7×C7⋊C3)⋊C3, C71(C3×C7⋊C3), SmallGroup(441,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C7⋊C32
 Chief series C1 — C7 — C72 — C7×C7⋊C3 — C7⋊C32
 Lower central C72 — C7⋊C32
 Upper central C1

Generators and relations for C7⋊C32
G = < a,b,c,d | a7=b3=c7=d3=1, bab-1=a4, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of C7⋊C32

 class 1 3A 3B 3C 3D 3E 3F 3G 3H 7A 7B 7C 7D 7E 7F 7G 7H 21A 21B 21C 21D 21E 21F 21G 21H size 1 7 7 7 7 49 49 49 49 3 3 3 3 9 9 9 9 21 21 21 21 21 21 21 21 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 linear of order 3 ρ3 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 1 ζ3 linear of order 3 ρ4 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 1 ζ32 linear of order 3 ρ5 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 1 1 1 ζ3 1 linear of order 3 ρ6 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ32 ζ3 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 linear of order 3 ρ7 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 1 1 1 ζ32 1 linear of order 3 ρ8 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ9 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ10 3 0 0 3 3 0 0 0 0 -1+√-7/2 3 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 0 0 0 -1+√-7/2 -1+√-7/2 -1-√-7/2 0 -1-√-7/2 complex lifted from C7⋊C3 ρ11 3 3 3 0 0 0 0 0 0 3 -1-√-7/2 -1+√-7/2 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 0 0 0 -1+√-7/2 0 complex lifted from C7⋊C3 ρ12 3 0 0 3 3 0 0 0 0 -1-√-7/2 3 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 0 0 0 -1-√-7/2 -1-√-7/2 -1+√-7/2 0 -1+√-7/2 complex lifted from C7⋊C3 ρ13 3 3 3 0 0 0 0 0 0 3 -1+√-7/2 -1-√-7/2 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 0 0 0 -1-√-7/2 0 complex lifted from C7⋊C3 ρ14 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3 -1-√-7/2 -1+√-7/2 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 0 0 0 ζ32ζ74+ζ32ζ72+ζ32ζ7 0 complex lifted from C3×C7⋊C3 ρ15 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3 -1+√-7/2 -1-√-7/2 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 0 0 0 ζ3ζ76+ζ3ζ75+ζ3ζ73 0 complex lifted from C3×C7⋊C3 ρ16 3 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1+√-7/2 3 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 0 0 0 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 0 ζ32ζ76+ζ32ζ75+ζ32ζ73 complex lifted from C3×C7⋊C3 ρ17 3 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1-√-7/2 3 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 0 0 0 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 0 ζ3ζ74+ζ3ζ72+ζ3ζ7 complex lifted from C3×C7⋊C3 ρ18 3 0 0 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1-√-7/2 3 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 0 0 0 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 0 ζ32ζ74+ζ32ζ72+ζ32ζ7 complex lifted from C3×C7⋊C3 ρ19 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 3 -1-√-7/2 -1+√-7/2 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 0 0 0 ζ3ζ74+ζ3ζ72+ζ3ζ7 0 complex lifted from C3×C7⋊C3 ρ20 3 0 0 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1+√-7/2 3 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 0 0 0 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 0 ζ3ζ76+ζ3ζ75+ζ3ζ73 complex lifted from C3×C7⋊C3 ρ21 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 3 -1+√-7/2 -1-√-7/2 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 0 0 0 ζ32ζ76+ζ32ζ75+ζ32ζ73 0 complex lifted from C3×C7⋊C3 ρ22 9 0 0 0 0 0 0 0 0 -3+3√-7/2 -3-3√-7/2 -3+3√-7/2 -3-3√-7/2 2 -3-√-7/2 -3+√-7/2 2 0 0 0 0 0 0 0 0 complex faithful ρ23 9 0 0 0 0 0 0 0 0 -3-3√-7/2 -3+3√-7/2 -3-3√-7/2 -3+3√-7/2 2 -3+√-7/2 -3-√-7/2 2 0 0 0 0 0 0 0 0 complex faithful ρ24 9 0 0 0 0 0 0 0 0 -3+3√-7/2 -3+3√-7/2 -3-3√-7/2 -3-3√-7/2 -3+√-7/2 2 2 -3-√-7/2 0 0 0 0 0 0 0 0 complex faithful ρ25 9 0 0 0 0 0 0 0 0 -3-3√-7/2 -3-3√-7/2 -3+3√-7/2 -3+3√-7/2 -3-√-7/2 2 2 -3+√-7/2 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C7⋊C32
On 21 points - transitive group 21T21
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 8 15)(2 10 19)(3 12 16)(4 14 20)(5 9 17)(6 11 21)(7 13 18)
(1 7 6 5 4 3 2)(8 13 11 9 14 12 10)(15 18 21 17 20 16 19)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)```

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

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

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

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

Matrix representation of C7⋊C32 in GL6(𝔽43)

 18 1 0 0 0 0 19 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 0 4 0 0 0 0 0 7 0 0 0 0 36 7 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 42 1 0 0 0 0 9 9 34 0 0 0 8 34 10
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 6 0 0 0 6 0 0 0 0 0 0 6 0

`G:=sub<GL(6,GF(43))| [18,19,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,0,36,0,0,0,4,7,7,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,9,8,0,0,0,1,9,34,0,0,0,0,34,10],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,6,0,0] >;`

C7⋊C32 in GAP, Magma, Sage, TeX

`C_7\rtimes C_3^2`
`% in TeX`

`G:=Group("C7:C3^2");`
`// GroupNames label`

`G:=SmallGroup(441,9);`
`// by ID`

`G=gap.SmallGroup(441,9);`
`# by ID`

`G:=PCGroup([4,-3,-3,-7,-7,78,2019]);`
`// Polycyclic`

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

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