Q₃₂ #113

order	32
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) = AGL(1,ℤ/16) (order 128) · Inn(G) = D₈ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = #72214802
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈
quaternion	Q₃₂

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂	D₈
commutator, Frattini, cosocle	C₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)

Polycyclic presentation:

2^5|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,c^b=cd,d^b=de

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁
size	1	1	2	2	2	2	2	2	2	8	8
order	1	2	4	8	8	16	16	16	16	4	4
χ₁	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	1	1	1	1	1	1	-1	-1
χ₃	1	1	1	1	1	-1	-1	-1	-1	1	-1
χ₄	1	1	1	1	1	-1	-1	-1	-1	-1	1
χ₅	2	2	2	-2	-2	0	0	0	0	0	0
χ₆	2	2	-2	0	0	$\sqrt{2}$	$\sqrt{2}$	$- \sqrt{2}$	$- \sqrt{2}$	0	0
χ₇	2	2	-2	0	0	$- \sqrt{2}$	$- \sqrt{2}$	$\sqrt{2}$	$\sqrt{2}$	0	0
χ₈	2	-2	0	$\sqrt{2}$	$- \sqrt{2}$	ζ−ζ⁷	−ζ+ζ⁷	ζ³−ζ⁵	−ζ³+ζ⁵	0	0
χ₉	2	-2	0	$\sqrt{2}$	$- \sqrt{2}$	−ζ+ζ⁷	ζ−ζ⁷	−ζ³+ζ⁵	ζ³−ζ⁵	0	0
χ₁₀	2	-2	0	$- \sqrt{2}$	$\sqrt{2}$	ζ³−ζ⁵	−ζ³+ζ⁵	−ζ+ζ⁷	ζ−ζ⁷	0	0
χ₁₁	2	-2	0	$- \sqrt{2}$	$\sqrt{2}$	−ζ³+ζ⁵	ζ³−ζ⁵	ζ−ζ⁷	−ζ+ζ⁷	0	0

Here ζ is a primitive root of unity of order 16 (ζ¹⁶ = 1).

Relationships

Fitting subgroup of	Q₃₂×A₅, #14515023
maximal subgroup of	SD₆₄, Q₆₄, C₂×Q₃₂, …
maximal normal subgroup of	SD₆₄, Q₆₄, C₂×Q₃₂, …
maximal quotient of	#358, #366, C₂×Q₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 20);

Q32 #113