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## G = C72⋊3C3order 147 = 3·72

### 3rd semidirect product of C72 and C3 acting faithfully

Aliases: C723C3, C72(C7⋊C3), SmallGroup(147,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C72⋊3C3
 Chief series C1 — C7 — C72 — C72⋊3C3
 Lower central C72 — C72⋊3C3
 Upper central C1

Generators and relations for C723C3
G = < a,b,c | a7=b7=c3=1, ab=ba, cac-1=a4, cbc-1=b2 >

Character table of C723C3

 class 1 3A 3B 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L 7M 7N 7O 7P size 1 49 49 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 3 0 0 -1-√-7/2 ζ76+2ζ74 ζ73+2ζ72 ζ75+ζ72+1 ζ75+2ζ7 ζ74+ζ73+1 ζ76+ζ7+1 -1+√-7/2 ζ75+ζ72+1 ζ76+ζ7+1 2ζ75+ζ74 ζ74+ζ73+1 2ζ73+ζ7 2ζ76+ζ72 -1-√-7/2 -1+√-7/2 complex faithful ρ5 3 0 0 -1-√-7/2 ζ73+2ζ72 ζ75+2ζ7 ζ76+ζ7+1 ζ76+2ζ74 ζ75+ζ72+1 ζ74+ζ73+1 -1+√-7/2 ζ76+ζ7+1 ζ74+ζ73+1 2ζ76+ζ72 ζ75+ζ72+1 2ζ75+ζ74 2ζ73+ζ7 -1-√-7/2 -1+√-7/2 complex faithful ρ6 3 0 0 -1-√-7/2 ζ74+ζ73+1 ζ75+ζ72+1 ζ76+2ζ74 ζ76+ζ7+1 ζ75+2ζ7 ζ73+2ζ72 -1+√-7/2 2ζ73+ζ7 2ζ75+ζ74 ζ75+ζ72+1 2ζ76+ζ72 ζ74+ζ73+1 ζ76+ζ7+1 -1+√-7/2 -1-√-7/2 complex faithful ρ7 3 0 0 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 3 3 complex lifted from C7⋊C3 ρ8 3 0 0 -1+√-7/2 ζ76+ζ7+1 ζ74+ζ73+1 2ζ76+ζ72 ζ75+ζ72+1 2ζ75+ζ74 2ζ73+ζ7 -1-√-7/2 ζ75+2ζ7 ζ76+2ζ74 ζ74+ζ73+1 ζ73+2ζ72 ζ76+ζ7+1 ζ75+ζ72+1 -1-√-7/2 -1+√-7/2 complex faithful ρ9 3 0 0 -1+√-7/2 2ζ75+ζ74 2ζ76+ζ72 ζ76+ζ7+1 2ζ73+ζ7 ζ75+ζ72+1 ζ74+ζ73+1 -1-√-7/2 ζ76+ζ7+1 ζ74+ζ73+1 ζ75+2ζ7 ζ75+ζ72+1 ζ73+2ζ72 ζ76+2ζ74 -1+√-7/2 -1-√-7/2 complex faithful ρ10 3 0 0 -1+√-7/2 2ζ73+ζ7 2ζ75+ζ74 ζ75+ζ72+1 2ζ76+ζ72 ζ74+ζ73+1 ζ76+ζ7+1 -1-√-7/2 ζ75+ζ72+1 ζ76+ζ7+1 ζ73+2ζ72 ζ74+ζ73+1 ζ76+2ζ74 ζ75+2ζ7 -1+√-7/2 -1-√-7/2 complex faithful ρ11 3 0 0 -1+√-7/2 ζ74+ζ73+1 ζ75+ζ72+1 2ζ73+ζ7 ζ76+ζ7+1 2ζ76+ζ72 2ζ75+ζ74 -1-√-7/2 ζ76+2ζ74 ζ73+2ζ72 ζ75+ζ72+1 ζ75+2ζ7 ζ74+ζ73+1 ζ76+ζ7+1 -1-√-7/2 -1+√-7/2 complex faithful ρ12 3 0 0 -1-√-7/2 ζ75+2ζ7 ζ76+2ζ74 ζ74+ζ73+1 ζ73+2ζ72 ζ76+ζ7+1 ζ75+ζ72+1 -1+√-7/2 ζ74+ζ73+1 ζ75+ζ72+1 2ζ73+ζ7 ζ76+ζ7+1 2ζ76+ζ72 2ζ75+ζ74 -1-√-7/2 -1+√-7/2 complex faithful ρ13 3 0 0 -1-√-7/2 ζ75+ζ72+1 ζ76+ζ7+1 ζ73+2ζ72 ζ74+ζ73+1 ζ76+2ζ74 ζ75+2ζ7 -1+√-7/2 2ζ75+ζ74 2ζ76+ζ72 ζ76+ζ7+1 2ζ73+ζ7 ζ75+ζ72+1 ζ74+ζ73+1 -1+√-7/2 -1-√-7/2 complex faithful ρ14 3 0 0 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -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 -1+√-7/2 complex lifted from C7⋊C3 ρ15 3 0 0 -1+√-7/2 2ζ76+ζ72 2ζ73+ζ7 ζ74+ζ73+1 2ζ75+ζ74 ζ76+ζ7+1 ζ75+ζ72+1 -1-√-7/2 ζ74+ζ73+1 ζ75+ζ72+1 ζ76+2ζ74 ζ76+ζ7+1 ζ75+2ζ7 ζ73+2ζ72 -1+√-7/2 -1-√-7/2 complex faithful ρ16 3 0 0 -1-√-7/2 ζ76+ζ7+1 ζ74+ζ73+1 ζ75+2ζ7 ζ75+ζ72+1 ζ73+2ζ72 ζ76+2ζ74 -1+√-7/2 2ζ76+ζ72 2ζ73+ζ7 ζ74+ζ73+1 2ζ75+ζ74 ζ76+ζ7+1 ζ75+ζ72+1 -1+√-7/2 -1-√-7/2 complex faithful ρ17 3 0 0 -1+√-7/2 ζ75+ζ72+1 ζ76+ζ7+1 2ζ75+ζ74 ζ74+ζ73+1 2ζ73+ζ7 2ζ76+ζ72 -1-√-7/2 ζ73+2ζ72 ζ75+2ζ7 ζ76+ζ7+1 ζ76+2ζ74 ζ75+ζ72+1 ζ74+ζ73+1 -1-√-7/2 -1+√-7/2 complex faithful ρ18 3 0 0 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -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 -1-√-7/2 complex lifted from C7⋊C3 ρ19 3 0 0 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1-√-7/2 3 3 complex lifted from C7⋊C3

Permutation representations of C723C3
On 21 points - transitive group 21T12
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 3 5 7 2 4 6)(8 12 9 13 10 14 11)(15 16 17 18 19 20 21)
(1 17 13)(2 19 10)(3 21 14)(4 16 11)(5 18 8)(6 20 12)(7 15 9)```

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

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

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

`G:=TransitiveGroup(21,12);`

C723C3 is a maximal subgroup of   C72⋊S3  C73F7  C72⋊C6  C7⋊C32
C723C3 is a maximal quotient of   C723C9

Matrix representation of C723C3 in GL3(𝔽43) generated by

 21 0 0 0 11 0 0 0 35
,
 41 0 0 0 16 0 0 0 4
,
 0 1 0 0 0 1 1 0 0
`G:=sub<GL(3,GF(43))| [21,0,0,0,11,0,0,0,35],[41,0,0,0,16,0,0,0,4],[0,0,1,1,0,0,0,1,0] >;`

C723C3 in GAP, Magma, Sage, TeX

`C_7^2\rtimes_3C_3`
`% in TeX`

`G:=Group("C7^2:3C3");`
`// GroupNames label`

`G:=SmallGroup(147,5);`
`// by ID`

`G=gap.SmallGroup(147,5);`
`# by ID`

`G:=PCGroup([3,-3,-7,-7,37,758]);`
`// Polycyclic`

`G:=Group<a,b,c|a^7=b^7=c^3=1,a*b=b*a,c*a*c^-1=a^4,c*b*c^-1=b^2>;`
`// generators/relations`

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