D7 - GroupNames

Permutation representations of D₇
►On 7 points: primitive - transitive group 7T2

Generators in S₇

(1 2 3 4 5 6 7)

(1 7)(2 6)(3 5)

G:=sub<Sym(7)| (1,2,3,4,5,6,7), (1,7)(2,6)(3,5)>;

G:=Group( (1,2,3,4,5,6,7), (1,7)(2,6)(3,5) );

G=PermutationGroup([[(1,2,3,4,5,6,7)], [(1,7),(2,6),(3,5)]])

G:=TransitiveGroup(7,2);

Generators in S₁₄

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

(1 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)

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

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

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

G:=TransitiveGroup(14,2);

Polynomial with Galois group D₇ over ℚ

action	f(x)	Disc(f)
7T2	x⁷-2x⁶-x⁵+x⁴+x³+x²-x-1	-71³
14T2	x¹⁴+9x¹²+53x¹⁰+333x⁸+1251x⁶+731x⁴+5415x²+8591	-2¹⁸²·7⁴·11²·71⁷

11	8
5	12

12	0
8	1

G:=sub<GL(2,GF(13))| [11,5,8,12],[12,8,0,1] >;

D₇ in GAP, Magma, Sage, TeX

D_7

% in TeX

G:=Group("D7");

// GroupNames label

G:=SmallGroup(14,1);

// by ID

G=gap.SmallGroup(14,1);

# by ID

G:=PCGroup([2,-2,-7,49]);

// Polycyclic

G:=Group<a,b|a^7=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₇ in TeX
Character table of D₇ in TeX

G = D7 order 14 = 2·7