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G = D72order 196 = 22·72

Direct product of D7 and D7

direct product, metabelian, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C72 — D72
C1C7C72C7×D7 — D72
C72 — D72
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 12A2B2C7A7B7C7D7E7F7G7H7I7J7K7L7M7N7O14A14B14C14D14E14F
 size 17749222222444444444141414141414
ρ11111111111111111111111111    trivial
ρ21-11-1111111111111111-1-1111-1    linear of order 2
ρ31-1-11111111111111111-1-1-1-1-1-1    linear of order 2
ρ411-1-111111111111111111-1-1-11    linear of order 2
ρ52-200222ζ7473ζ767ζ7572ζ767ζ7572ζ767ζ7473ζ767ζ7572ζ7572ζ7473ζ747375727473000767    orthogonal lifted from D14
ρ62-200222ζ767ζ7572ζ7473ζ7572ζ7473ζ7572ζ767ζ7572ζ7473ζ7473ζ767ζ76774737670007572    orthogonal lifted from D14
ρ72-200222ζ7572ζ7473ζ767ζ7473ζ767ζ7473ζ7572ζ7473ζ767ζ767ζ7572ζ757276775720007473    orthogonal lifted from D14
ρ820-20ζ7473ζ767ζ7572222ζ767ζ767ζ7473ζ7572ζ7572ζ7572ζ7473ζ7473ζ76700747376775720    orthogonal lifted from D14
ρ920-20ζ767ζ7572ζ7473222ζ7572ζ7572ζ767ζ7473ζ7473ζ7473ζ767ζ767ζ757200767757274730    orthogonal lifted from D14
ρ102200222ζ7572ζ7473ζ767ζ7473ζ767ζ7473ζ7572ζ7473ζ767ζ767ζ7572ζ7572ζ767ζ7572000ζ7473    orthogonal lifted from D7
ρ1120-20ζ7572ζ7473ζ767222ζ7473ζ7473ζ7572ζ767ζ767ζ767ζ7572ζ7572ζ747300757274737670    orthogonal lifted from D14
ρ122020ζ767ζ7572ζ7473222ζ7572ζ7572ζ767ζ7473ζ7473ζ7473ζ767ζ767ζ757200ζ767ζ7572ζ74730    orthogonal lifted from D7
ρ132020ζ7572ζ7473ζ767222ζ7473ζ7473ζ7572ζ767ζ767ζ767ζ7572ζ7572ζ747300ζ7572ζ7473ζ7670    orthogonal lifted from D7
ρ142020ζ7473ζ767ζ7572222ζ767ζ767ζ7473ζ7572ζ7572ζ7572ζ7473ζ7473ζ76700ζ7473ζ767ζ75720    orthogonal lifted from D7
ρ152200222ζ7473ζ767ζ7572ζ767ζ7572ζ767ζ7473ζ767ζ7572ζ7572ζ7473ζ7473ζ7572ζ7473000ζ767    orthogonal lifted from D7
ρ162200222ζ767ζ7572ζ7473ζ7572ζ7473ζ7572ζ767ζ7572ζ7473ζ7473ζ767ζ767ζ7473ζ767000ζ7572    orthogonal lifted from D7
ρ17400076+2ζ775+2ζ7274+2ζ7374+2ζ7376+2ζ775+2ζ727572-1ζ7473+2ζ7572+2ζ767+2767-17473-17572-1767-17473-1000000    orthogonal faithful
ρ18400074+2ζ7376+2ζ775+2ζ7275+2ζ7274+2ζ7376+2ζ7767-1ζ7572+2ζ767+2ζ7473+27473-17572-1767-17473-17572-1000000    orthogonal faithful
ρ19400074+2ζ7376+2ζ775+2ζ7274+2ζ7376+2ζ775+2ζ72ζ7572+27572-1767-17473-17572-1ζ7473+27473-1ζ767+2767-1000000    orthogonal faithful
ρ20400074+2ζ7376+2ζ775+2ζ7276+2ζ775+2ζ7274+2ζ737572-1767-17473-17572-1ζ7473+27473-1ζ767+2767-1ζ7572+2000000    orthogonal faithful
ρ21400076+2ζ775+2ζ7274+2ζ7375+2ζ7274+2ζ7376+2ζ77473-17572-1767-17473-1ζ767+2767-1ζ7572+27572-1ζ7473+2000000    orthogonal faithful
ρ22400075+2ζ7274+2ζ7376+2ζ774+2ζ7376+2ζ775+2ζ72767-17473-17572-1767-1ζ7572+27572-1ζ7473+27473-1ζ767+2000000    orthogonal faithful
ρ23400075+2ζ7274+2ζ7376+2ζ776+2ζ775+2ζ7274+2ζ737473-1ζ767+2ζ7473+2ζ7572+27572-1767-17473-17572-1767-1000000    orthogonal faithful
ρ24400075+2ζ7274+2ζ7376+2ζ775+2ζ7274+2ζ7376+2ζ7ζ767+2767-17473-17572-1767-1ζ7572+27572-1ζ7473+27473-1000000    orthogonal faithful
ρ25400076+2ζ775+2ζ7274+2ζ7376+2ζ775+2ζ7274+2ζ73ζ7473+27473-17572-1767-17473-1ζ767+2767-1ζ7572+27572-1000000    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

1000
0100
0074
0070
,
28000
02800
001911
002010
,
101800
172200
0010
0001
,
28100
0100
00280
00028
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

Export

Subgroup lattice of D72 in TeX
Character table of D72 in TeX

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