Dic₅₁₆₃ #35207657

order	20652
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₁₆₃
unambiguous semidirect product	C₃⋊Dic₁₇₂₁

Splittings

derived, Hall	C₃×C₁₇₂₁ ⋊ C₄
Sylow	C₁₇₂₁ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₇₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₁₆₃
commutator, hypocenter	C₃×C₁₇₂₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₇₂₁	C₂
LCS	G ▷ C₃×C₁₇₂₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1728
transitive degree	μ_t(G) = 20652

A minimal faithful permutation representation — 2 generators on 1728 points:

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Polycyclic presentation:

2,3,1721,2|a^2=d,b^a=b^2,c^a=c^1720

Conjugacy classes & character table

5166 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₃₂₆, Dic₁₅₄₈₉
maximal normal subgroup of	Dic₁₀₃₂₆
maximal quotient of	Dic₁₅₄₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20652, 4);

Dic5163 #35207657