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## G = D72order 196 = 22·72

### Direct product of D7 and D7

Aliases: D72, C71D14, C72⋊C22, C7⋊D7⋊C2, (C7×D7)⋊C2, SmallGroup(196,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — D72
 Chief series C1 — C7 — C72 — C7×D7 — D72
 Lower central C72 — D72
 Upper central C1

Generators and relations for D72
G = < a,b,c,d | a7=b2=c7=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

7C2
7C2
49C2
2C7
2C7
2C7
49C22
7C14
7D7
7C14
7D7
14D7
14D7
14D7
7D14
7D14

Character table of D72

 class 1 2A 2B 2C 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K 7L 7M 7N 7O 14A 14B 14C 14D 14E 14F size 1 7 7 49 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 14 14 14 14 14 14 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 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 linear of order 2 ρ3 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 linear of order 2 ρ4 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 linear of order 2 ρ5 2 -2 0 0 2 2 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 0 0 0 -ζ76-ζ7 orthogonal lifted from D14 ρ6 2 -2 0 0 2 2 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 0 0 0 -ζ75-ζ72 orthogonal lifted from D14 ρ7 2 -2 0 0 2 2 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 0 0 0 -ζ74-ζ73 orthogonal lifted from D14 ρ8 2 0 -2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 2 2 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 0 orthogonal lifted from D14 ρ9 2 0 -2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 2 2 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 0 orthogonal lifted from D14 ρ10 2 2 0 0 2 2 2 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 0 0 0 ζ74+ζ73 orthogonal lifted from D7 ρ11 2 0 -2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 2 2 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 0 orthogonal lifted from D14 ρ12 2 0 2 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 2 2 2 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 0 orthogonal lifted from D7 ρ13 2 0 2 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 2 2 2 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 0 orthogonal lifted from D7 ρ14 2 0 2 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 2 2 2 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 0 orthogonal lifted from D7 ρ15 2 2 0 0 2 2 2 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 0 0 0 ζ76+ζ7 orthogonal lifted from D7 ρ16 2 2 0 0 2 2 2 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 0 0 0 ζ75+ζ72 orthogonal lifted from D7 ρ17 4 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -ζ75-ζ72-1 ζ74+ζ73+2 ζ75+ζ72+2 ζ76+ζ7+2 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 0 0 0 0 0 0 orthogonal faithful ρ18 4 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -ζ76-ζ7-1 ζ75+ζ72+2 ζ76+ζ7+2 ζ74+ζ73+2 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 0 0 0 0 0 0 orthogonal faithful ρ19 4 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 ζ75+ζ72+2 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 ζ74+ζ73+2 -ζ74-ζ73-1 ζ76+ζ7+2 -ζ76-ζ7-1 0 0 0 0 0 0 orthogonal faithful ρ20 4 0 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 ζ74+ζ73+2 -ζ74-ζ73-1 ζ76+ζ7+2 -ζ76-ζ7-1 ζ75+ζ72+2 0 0 0 0 0 0 orthogonal faithful ρ21 4 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 ζ76+ζ7+2 -ζ76-ζ7-1 ζ75+ζ72+2 -ζ75-ζ72-1 ζ74+ζ73+2 0 0 0 0 0 0 orthogonal faithful ρ22 4 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 ζ75+ζ72+2 -ζ75-ζ72-1 ζ74+ζ73+2 -ζ74-ζ73-1 ζ76+ζ7+2 0 0 0 0 0 0 orthogonal faithful ρ23 4 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -ζ74-ζ73-1 ζ76+ζ7+2 ζ74+ζ73+2 ζ75+ζ72+2 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 0 0 0 0 0 0 orthogonal faithful ρ24 4 0 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 ζ76+ζ7+2 -ζ76-ζ7-1 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 ζ75+ζ72+2 -ζ75-ζ72-1 ζ74+ζ73+2 -ζ74-ζ73-1 0 0 0 0 0 0 orthogonal faithful ρ25 4 0 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 ζ74+ζ73+2 -ζ74-ζ73-1 -ζ75-ζ72-1 -ζ76-ζ7-1 -ζ74-ζ73-1 ζ76+ζ7+2 -ζ76-ζ7-1 ζ75+ζ72+2 -ζ75-ζ72-1 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D72
On 14 points - transitive group 14T13
Generators in S14
```(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)
(1 3 5 7 2 4 6)(8 13 11 9 14 12 10)
(1 11)(2 12)(3 13)(4 14)(5 8)(6 9)(7 10)```

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

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

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

`G:=TransitiveGroup(14,13);`

On 28 points - transitive group 28T36
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 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 26)(16 25)(17 24)(18 23)(19 22)(20 28)(21 27)
(1 3 5 7 2 4 6)(8 13 11 9 14 12 10)(15 20 18 16 21 19 17)(22 24 26 28 23 25 27)
(1 21)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)```

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

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

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

`G:=TransitiveGroup(28,36);`

D72 is a maximal subgroup of   D7≀C2
D72 is a maximal quotient of   Dic72D7  C722D4  C7⋊D28  C722Q8

Polynomial with Galois group D72 over ℚ
actionf(x)Disc(f)
14T13x14-2x13-29x12-222x11-352x10+3498x9+18163x8+46467x7+92188x6+128405x5+96637x4+31142x3+7064x2+6304x+2432236·74·232·7111·1517·14232·17232·1610392·17685172

Matrix representation of D72 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 7 4 0 0 7 0
,
 28 0 0 0 0 28 0 0 0 0 19 11 0 0 20 10
,
 10 18 0 0 17 22 0 0 0 0 1 0 0 0 0 1
,
 28 1 0 0 0 1 0 0 0 0 28 0 0 0 0 28
`G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,7,7,0,0,4,0],[28,0,0,0,0,28,0,0,0,0,19,20,0,0,11,10],[10,17,0,0,18,22,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,1,1,0,0,0,0,28,0,0,0,0,28] >;`

D72 in GAP, Magma, Sage, TeX

`D_7^2`
`% in TeX`

`G:=Group("D7^2");`
`// GroupNames label`

`G:=SmallGroup(196,9);`
`// by ID`

`G=gap.SmallGroup(196,9);`
`# by ID`

`G:=PCGroup([4,-2,-2,-7,-7,150,2691]);`
`// Polycyclic`

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

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