SD32 - GroupNames

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G = SD₃₂ order 32 = 2⁵

Semidihedral group

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

Aliases: SD₃₂, D₈.C₂, C₁₆⋊₂C₂, C₄.₂D₄, C₂.₄D₈, Q₁₆⋊₁C₂, C₈.₃C₂², 2-Sylow(GL(2,7)), also known as the quasi-dihedral group QD₃₂, SmallGroup(32,19)

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

C₁ — C₈ — SD₃₂

C₁ — C₂ — C₄ — C₈ — D₈ — SD₃₂

C₁ — C₂ — C₄ — C₈ — SD₃₂

C₁ — C₂ — C₄ — C₈ — SD₃₂

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₈ — SD₃₂

Generators and relations for SD₃₂
G = < a,b | a¹⁶=b²=1, bab=a⁷ >

C₁

C₂

₈C₂

C₄

₄C₄

₄C₂²

C₈

₂Q₈

₂D₄

Q₁₆

C₁₆

D₈

SD₃₂

Character table of SD₃₂

class	1	2A	2B	4A	4B	8A	8B	16A	16B	16C	16D
size	1	1	8	2	8	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	2	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₆	2	2	0	-2	0	0	0	√2	-√2	√2	-√2	orthogonal lifted from D₈
ρ₇	2	2	0	-2	0	0	0	-√2	√2	-√2	√2	orthogonal lifted from D₈
ρ₈	2	-2	0	0	0	-√2	√2	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	complex faithful
ρ₉	2	-2	0	0	0	√2	-√2	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	complex faithful
ρ₁₀	2	-2	0	0	0	-√2	√2	ζ₁₆¹³+ζ₁₆¹¹	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	complex faithful
ρ₁₁	2	-2	0	0	0	√2	-√2	ζ₁₆¹⁵+ζ₁₆⁹	ζ₁₆⁵+ζ₁₆³	ζ₁₆⁷+ζ₁₆	ζ₁₆¹³+ζ₁₆¹¹	complex faithful

Permutation representations of SD₃₂
►On 16 points - transitive group 16T55

Generators in S₁₆

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

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

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

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

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

G:=TransitiveGroup(16,55);

SD₃₂ is a maximal subgroup of
C₃²⋊SD₃₂
C₈.D_2p: C₄○D₁₆ C₁₆⋊C₂² Q₃₂⋊C₂ C₄₈⋊C₂ D₈.S₃ C₈.₆D₆ C₁₆⋊D₅ D₈.D₅ ...
SD₃₂ is a maximal quotient of
C₁₆⋊₄C₄ C₃²⋊SD₃₂
C₈.D_2p: C₂.D₁₆ C₂.Q₃₂ C₄₈⋊C₂ D₈.S₃ C₈.₆D₆ C₁₆⋊D₅ D₈.D₅ C₅⋊SD₃₂ ...

Matrix representation of SD₃₂ ►in GL₂(𝔽₇) generated by

0	1
1	6

,

1	6
0	6

G:=sub<GL(2,GF(7))| [0,1,1,6],[1,0,6,6] >;

SD₃₂ in GAP, Magma, Sage, TeX

{\rm SD}_{32}

% in TeX

G:=Group("SD32");

// GroupNames label

G:=SmallGroup(32,19);

// by ID

G=gap.SmallGroup(32,19);

# by ID

G:=PCGroup([5,-2,2,-2,-2,-2,80,61,182,97,102,483,248,58]);

// Polycyclic

G:=Group<a,b|a^16=b^2=1,b*a*b=a^7>;

// generators/relations

Export

Subgroup lattice of SD₃₂ in TeX
Character table of SD₃₂ in TeX

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