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## G = C72⋊C3order 147 = 3·72

### 2nd semidirect product of C72 and C3 acting faithfully

Aliases: C722C3, C71(C7⋊C3), SmallGroup(147,4)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C72⋊C3
G = < a,b,c | a7=b7=c3=1, ab=ba, cac-1=a4, cbc-1=b4 >

Character table of C72⋊C3

 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 -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 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ5 3 0 0 -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 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ6 3 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 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 complex lifted from C7⋊C3 ρ7 3 0 0 -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 3 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ8 3 0 0 -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 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ9 3 0 0 -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 -1-√-7/2 -1-√-7/2 3 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ10 3 0 0 -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 3 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ11 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 ρ12 3 0 0 -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 3 -1-√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ13 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 ρ14 3 0 0 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 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 complex lifted from C7⋊C3 ρ15 3 0 0 -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 -1+√-7/2 -1+√-7/2 3 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ16 3 0 0 -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 3 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ17 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 ρ18 3 0 0 -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 3 -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

Smallest permutation representation of C72⋊C3
On 49 points
Generators in S49
```(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)
(1 49 27 34 41 13 20)(2 43 28 35 42 14 21)(3 44 22 29 36 8 15)(4 45 23 30 37 9 16)(5 46 24 31 38 10 17)(6 47 25 32 39 11 18)(7 48 26 33 40 12 19)
(2 3 5)(4 7 6)(8 31 21)(9 33 18)(10 35 15)(11 30 19)(12 32 16)(13 34 20)(14 29 17)(22 38 43)(23 40 47)(24 42 44)(25 37 48)(26 39 45)(27 41 49)(28 36 46)```

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

`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)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49), (1,49,27,34,41,13,20)(2,43,28,35,42,14,21)(3,44,22,29,36,8,15)(4,45,23,30,37,9,16)(5,46,24,31,38,10,17)(6,47,25,32,39,11,18)(7,48,26,33,40,12,19), (2,3,5)(4,7,6)(8,31,21)(9,33,18)(10,35,15)(11,30,19)(12,32,16)(13,34,20)(14,29,17)(22,38,43)(23,40,47)(24,42,44)(25,37,48)(26,39,45)(27,41,49)(28,36,46) );`

`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),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49)], [(1,49,27,34,41,13,20),(2,43,28,35,42,14,21),(3,44,22,29,36,8,15),(4,45,23,30,37,9,16),(5,46,24,31,38,10,17),(6,47,25,32,39,11,18),(7,48,26,33,40,12,19)], [(2,3,5),(4,7,6),(8,31,21),(9,33,18),(10,35,15),(11,30,19),(12,32,16),(13,34,20),(14,29,17),(22,38,43),(23,40,47),(24,42,44),(25,37,48),(26,39,45),(27,41,49),(28,36,46)]])`

C72⋊C3 is a maximal subgroup of   C74F7  C7⋊F7  C7⋊C32
C72⋊C3 is a maximal quotient of   C72⋊C9

Matrix representation of C72⋊C3 in GL6(𝔽43)

 7 37 0 0 0 0 9 36 1 0 0 0 37 24 18 0 0 0 0 0 0 1 25 24 0 0 0 24 42 42 0 0 0 42 42 18
,
 1 0 0 0 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 1 25 24
,
 1 0 0 0 0 0 20 42 42 0 0 0 41 1 0 0 0 0 0 0 0 1 0 0 0 0 0 24 42 42 0 0 0 0 1 0

`G:=sub<GL(6,GF(43))| [7,9,37,0,0,0,37,36,24,0,0,0,0,1,18,0,0,0,0,0,0,1,24,42,0,0,0,25,42,42,0,0,0,24,42,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,25,0,0,0,0,1,24],[1,20,41,0,0,0,0,42,1,0,0,0,0,42,0,0,0,0,0,0,0,1,24,0,0,0,0,0,42,1,0,0,0,0,42,0] >;`

C72⋊C3 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(147,4);`
`// by ID`

`G=gap.SmallGroup(147,4);`
`# by ID`

`G:=PCGroup([3,-3,-7,-7,37,380]);`
`// 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^4>;`
`// generators/relations`

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