D13 - GroupNames

class	1	2	13A	13B	13C	13D	13E	13F
size	1	13	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	linear of order 2
ρ₃	2	0	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃¹²+ζ₁₃	ζ₁₃¹¹+ζ₁₃²	orthogonal faithful
ρ₄	2	0	ζ₁₃¹²+ζ₁₃	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃⁷+ζ₁₃⁶	orthogonal faithful
ρ₅	2	0	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃¹²+ζ₁₃	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃⁸+ζ₁₃⁵	orthogonal faithful
ρ₆	2	0	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃¹²+ζ₁₃	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹⁰+ζ₁₃³	orthogonal faithful
ρ₇	2	0	ζ₁₃¹¹+ζ₁₃²	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃⁹+ζ₁₃⁴	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃¹²+ζ₁₃	orthogonal faithful
ρ₈	2	0	ζ₁₃⁸+ζ₁₃⁵	ζ₁₃¹²+ζ₁₃	ζ₁₃¹⁰+ζ₁₃³	ζ₁₃⁷+ζ₁₃⁶	ζ₁₃¹¹+ζ₁₃²	ζ₁₃⁹+ζ₁₃⁴	orthogonal faithful

Permutation representations of D₁₃
►On 13 points: primitive - transitive group 13T2

Generators in S₁₃

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

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

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

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

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

G:=TransitiveGroup(13,2);

►Regular action on 26 points - transitive group 26T2

Generators in S₂₆

(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)

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

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

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

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

G:=TransitiveGroup(26,2);

D₁₃ is a maximal subgroup of
C₁₃⋊C₄ C₁₃⋊C₆ C₁₃⋊D₁₃
D_13p: D₃₉ D₆₅ D₉₁ D₁₄₃ D₁₆₉ D₂₂₁ D₂₄₇ ...
D₁₃ is a maximal quotient of
Dic₁₃ C₁₃⋊D₁₃
D_13p: D₃₉ D₆₅ D₉₁ D₁₄₃ D₁₆₉ D₂₂₁ D₂₄₇ ...

Polynomial with Galois group D₁₃ over ℚ

action	f(x)	Disc(f)
13T2	x¹³-x¹²-50x¹¹-6x¹⁰+857x⁹+943x⁸-5045x⁷-9319x⁶+3890x⁵+13442x⁴+1835x³-2759x²+304x+4	2⁶⁴·41²·53²·71²·149²·8101⁶

Matrix representation of D₁₃ ►in GL₂(𝔽₅₃) generated by

27	52
28	52

41	26
21	12

G:=sub<GL(2,GF(53))| [27,28,52,52],[41,21,26,12] >;

D₁₃ in GAP, Magma, Sage, TeX

D_{13}

% in TeX

G:=Group("D13");

// GroupNames label

G:=SmallGroup(26,1);

// by ID

G=gap.SmallGroup(26,1);

# by ID

G:=PCGroup([2,-2,-13,97]);

// Polycyclic

G:=Group<a,b|a^13=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₁₃ in TeX
Character table of D₁₃ in TeX

G = D13 order 26 = 2·13

Dihedral group

G = D₁₃ order 26 = 2·13