D16 - GroupNames

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G = D₁₆ order 32 = 2⁵

Dihedral group

p-group, metacyclic, nilpotent (class 4), monomial

Aliases: D₁₆, C₁₆⋊₁C₂, D₈⋊₁C₂, C₂.₃D₈, C₄.₁D₄, C₈.₂C₂², 2-Sylow(PGL(2,17)), sometimes denoted D₃₂ or Dih₁₆ or Dih₃₂, SmallGroup(32,18)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

C₁ — C₈ — D₁₆

C₁ — C₂ — C₄ — C₈ — D₈ — D₁₆

C₁ — C₂ — C₄ — C₈ — D₁₆

C₁ — C₂ — C₄ — C₈ — D₁₆

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₈ — D₁₆

Generators and relations for D₁₆
G = < a,b | a¹⁶=b²=1, bab=a^-1 >

C₁

C₂

₈C₂

₈C₂

C₄

₄C₂²

₄C₂²

C₈

₂D₄

₂D₄

C₁₆

D₈

D₈

D₁₆

Character table of D₁₆

class	1	2A	2B	2C	4	8A	8B	16A	16B	16C	16D
size	1	1	8	8	2	2	2	2	2	2	2
ρ₁	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	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	linear of order 2
ρ₄	1	1	-1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₅	2	2	0	0	2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₆	2	2	0	0	-2	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₇	2	2	0	0	-2	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₈	2	-2	0	0	0	-√2	√2	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	orthogonal faithful
ρ₉	2	-2	0	0	0	-√2	√2	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	orthogonal faithful
ρ₁₀	2	-2	0	0	0	√2	-√2	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁷+ζ₁₆	ζ₁₆⁷-ζ₁₆	orthogonal faithful
ρ₁₁	2	-2	0	0	0	√2	-√2	ζ₁₆⁵-ζ₁₆³	-ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷-ζ₁₆	-ζ₁₆⁷+ζ₁₆	orthogonal faithful

Permutation representations of D₁₆
►On 16 points - transitive group 16T56

Generators in S₁₆

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]])

G:=TransitiveGroup(16,56);

D₁₆ is a maximal subgroup of
SD₆₄ C₃²⋊D₁₆
D_16p: D₃₂ D₄₈ D₈₀ D₁₁₂ D₁₇₆ D₂₀₈ ...
D_8p⋊C₂: C₄○D₁₆ C₁₆⋊C₂² C₃⋊D₁₆ C₅⋊D₁₆ C₇⋊D₁₆ C₁₁⋊D₁₆ C₁₃⋊D₁₆ ...
D₁₆ is a maximal quotient of
C₁₆⋊₃C₄ C₃²⋊D₁₆
D_16p: D₃₂ D₄₈ D₈₀ D₁₁₂ D₁₇₆ D₂₀₈ ...
C_4p.D₄: C₂.D₁₆ SD₆₄ Q₆₄ C₃⋊D₁₆ C₅⋊D₁₆ C₇⋊D₁₆ C₁₁⋊D₁₆ C₁₃⋊D₁₆ ...

Matrix representation of D₁₆ ►in GL₂(𝔽₁₇) generated by

11	4
13	11

,

1	0
0	16

G:=sub<GL(2,GF(17))| [11,13,4,11],[1,0,0,16] >;

D₁₆ in GAP, Magma, Sage, TeX

D_{16}

% in TeX

G:=Group("D16");

// GroupNames label

G:=SmallGroup(32,18);

// by ID

G=gap.SmallGroup(32,18);

# by ID

G:=PCGroup([5,-2,2,-2,-2,-2,61,182,97,102,483,248,58]);

// Polycyclic

G:=Group<a,b|a^16=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₁₆ in TeX
Character table of D₁₆ in TeX

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