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## G = C72⋊C4order 196 = 22·72

### The semidirect product of C72 and C4 acting faithfully

Aliases: C72⋊C4, C7⋊D7.C2, SmallGroup(196,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C72⋊C4
 Chief series C1 — C72 — C7⋊D7 — C72⋊C4
 Lower central C72 — C72⋊C4
 Upper central C1

Generators and relations for C72⋊C4
G = < a,b,c | a7=b7=c4=1, ab=ba, cac-1=a3b-1, cbc-1=a3b4 >

49C2
2C7
2C7
2C7
2C7
49C4
14D7
14D7
14D7
14D7

Character table of C72⋊C4

 class 1 2 4A 4B 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L size 1 49 49 49 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 i -i 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 4 0 0 0 -ζ74-ζ73-1 -ζ74-ζ73-1 -ζ76-ζ7-1 2ζ75+2ζ72 -ζ76-ζ7-1 2ζ74+2ζ73 -ζ75-ζ72-1 -ζ75-ζ72-1 2ζ76+2ζ7 ζ75+ζ72+2 ζ74+ζ73+2 ζ76+ζ7+2 orthogonal faithful ρ6 4 0 0 0 2ζ74+2ζ73 ζ75+ζ72+2 ζ74+ζ73+2 -ζ74-ζ73-1 2ζ76+2ζ7 -ζ76-ζ7-1 ζ76+ζ7+2 2ζ75+2ζ72 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73-1 orthogonal faithful ρ7 4 0 0 0 -ζ76-ζ7-1 -ζ76-ζ7-1 -ζ75-ζ72-1 ζ75+ζ72+2 -ζ75-ζ72-1 ζ74+ζ73+2 -ζ74-ζ73-1 -ζ74-ζ73-1 ζ76+ζ7+2 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 orthogonal faithful ρ8 4 0 0 0 -ζ74-ζ73-1 -ζ74-ζ73-1 -ζ76-ζ7-1 ζ76+ζ7+2 -ζ76-ζ7-1 ζ75+ζ72+2 -ζ75-ζ72-1 -ζ75-ζ72-1 ζ74+ζ73+2 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 orthogonal faithful ρ9 4 0 0 0 2ζ76+2ζ7 ζ74+ζ73+2 ζ76+ζ7+2 -ζ76-ζ7-1 2ζ75+2ζ72 -ζ75-ζ72-1 ζ75+ζ72+2 2ζ74+2ζ73 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7-1 orthogonal faithful ρ10 4 0 0 0 ζ76+ζ7+2 2ζ75+2ζ72 2ζ74+2ζ73 -ζ75-ζ72-1 ζ75+ζ72+2 -ζ74-ζ73-1 2ζ76+2ζ7 ζ74+ζ73+2 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72-1 orthogonal faithful ρ11 4 0 0 0 -ζ76-ζ7-1 -ζ76-ζ7-1 -ζ75-ζ72-1 2ζ74+2ζ73 -ζ75-ζ72-1 2ζ76+2ζ7 -ζ74-ζ73-1 -ζ74-ζ73-1 2ζ75+2ζ72 ζ74+ζ73+2 ζ76+ζ7+2 ζ75+ζ72+2 orthogonal faithful ρ12 4 0 0 0 ζ74+ζ73+2 2ζ76+2ζ7 2ζ75+2ζ72 -ζ76-ζ7-1 ζ76+ζ7+2 -ζ75-ζ72-1 2ζ74+2ζ73 ζ75+ζ72+2 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ74-ζ73-1 -ζ76-ζ7-1 orthogonal faithful ρ13 4 0 0 0 2ζ75+2ζ72 ζ76+ζ7+2 ζ75+ζ72+2 -ζ75-ζ72-1 2ζ74+2ζ73 -ζ74-ζ73-1 ζ74+ζ73+2 2ζ76+2ζ7 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ76-ζ7-1 -ζ75-ζ72-1 orthogonal faithful ρ14 4 0 0 0 ζ75+ζ72+2 2ζ74+2ζ73 2ζ76+2ζ7 -ζ74-ζ73-1 ζ74+ζ73+2 -ζ76-ζ7-1 2ζ75+2ζ72 ζ76+ζ7+2 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ75-ζ72-1 -ζ74-ζ73-1 orthogonal faithful ρ15 4 0 0 0 -ζ75-ζ72-1 -ζ75-ζ72-1 -ζ74-ζ73-1 2ζ76+2ζ7 -ζ74-ζ73-1 2ζ75+2ζ72 -ζ76-ζ7-1 -ζ76-ζ7-1 2ζ74+2ζ73 ζ76+ζ7+2 ζ75+ζ72+2 ζ74+ζ73+2 orthogonal faithful ρ16 4 0 0 0 -ζ75-ζ72-1 -ζ75-ζ72-1 -ζ74-ζ73-1 ζ74+ζ73+2 -ζ74-ζ73-1 ζ76+ζ7+2 -ζ76-ζ7-1 -ζ76-ζ7-1 ζ75+ζ72+2 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 orthogonal faithful

Permutation representations of C72⋊C4
On 14 points - transitive group 14T12
Generators in S14
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 6 4 2 7 5 3)(8 14 13 12 11 10 9)
(1 8)(2 11 7 12)(3 14 6 9)(4 10 5 13)```

`G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,6,4,2,7,5,3)(8,14,13,12,11,10,9), (1,8)(2,11,7,12)(3,14,6,9)(4,10,5,13)>;`

`G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,6,4,2,7,5,3)(8,14,13,12,11,10,9), (1,8)(2,11,7,12)(3,14,6,9)(4,10,5,13) );`

`G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,6,4,2,7,5,3),(8,14,13,12,11,10,9)], [(1,8),(2,11,7,12),(3,14,6,9),(4,10,5,13)])`

`G:=TransitiveGroup(14,12);`

On 28 points - transitive group 28T35
Generators in S28
```(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)
(1 6 4 2 7 5 3)(8 13 11 9 14 12 10)(15 21 20 19 18 17 16)(22 28 27 26 25 24 23)
(1 27 12 15)(2 23 11 19)(3 26 10 16)(4 22 9 20)(5 25 8 17)(6 28 14 21)(7 24 13 18)```

`G:=sub<Sym(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), (1,6,4,2,7,5,3)(8,13,11,9,14,12,10)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,27,12,15)(2,23,11,19)(3,26,10,16)(4,22,9,20)(5,25,8,17)(6,28,14,21)(7,24,13,18)>;`

`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), (1,6,4,2,7,5,3)(8,13,11,9,14,12,10)(15,21,20,19,18,17,16)(22,28,27,26,25,24,23), (1,27,12,15)(2,23,11,19)(3,26,10,16)(4,22,9,20)(5,25,8,17)(6,28,14,21)(7,24,13,18) );`

`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)], [(1,6,4,2,7,5,3),(8,13,11,9,14,12,10),(15,21,20,19,18,17,16),(22,28,27,26,25,24,23)], [(1,27,12,15),(2,23,11,19),(3,26,10,16),(4,22,9,20),(5,25,8,17),(6,28,14,21),(7,24,13,18)])`

`G:=TransitiveGroup(28,35);`

C72⋊C4 is a maximal subgroup of   C72⋊C8  D7≀C2  C72⋊Q8
C72⋊C4 is a maximal quotient of   C722C8

Polynomial with Galois group C72⋊C4 over ℚ
actionf(x)Disc(f)
14T12x14+28x12-189x11+756x10-4004x9+15953x8-48856x7+129262x6-251559x5+330764x4-272986x3-123305x2+739662x-57791622·518·738·192·1732·7012·21112·171592·9818430376572

Matrix representation of C72⋊C4 in GL4(𝔽29) generated by

 10 1 0 0 27 26 0 0 20 27 4 10 14 1 14 28
,
 1 0 0 0 0 1 0 0 21 27 4 10 4 1 14 28
,
 0 0 10 1 4 1 23 25 25 8 28 0 17 26 10 0
`G:=sub<GL(4,GF(29))| [10,27,20,14,1,26,27,1,0,0,4,14,0,0,10,28],[1,0,21,4,0,1,27,1,0,0,4,14,0,0,10,28],[0,4,25,17,0,1,8,26,10,23,28,10,1,25,0,0] >;`

C72⋊C4 in GAP, Magma, Sage, TeX

`C_7^2\rtimes C_4`
`% in TeX`

`G:=Group("C7^2:C4");`
`// GroupNames label`

`G:=SmallGroup(196,8);`
`// by ID`

`G=gap.SmallGroup(196,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-7,7,8,530,150,1603,1351]);`
`// Polycyclic`

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

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