Dic₅₀₉₇ #35205128

order	20388
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₀₉₇
unambiguous semidirect product	C₃⋊Dic₁₆₉₉
unambiguous semidirect product	C₁₆₉₉⋊Dic₃

Splittings

derived, Hall	C₃×C₁₆₉₉ ⋊ C₄
Sylow	C₁₆₉₉ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₆₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₀₉₇
commutator, hypocenter	C₃×C₁₆₉₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₆₉₉	C₂
LCS	G ▷ C₃×C₁₆₉₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1706
transitive degree	μ_t(G) = 20388

A minimal faithful permutation representation — 2 generators on 1706 points:

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Polycyclic presentation:

2,3,1699,2|a^2=d,b^a=b^2,c^a=c^1698

Conjugacy classes & character table

5100 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₀₁₉₄, Dic₁₅₂₉₁
maximal normal subgroup of	Dic₁₀₁₉₄
maximal quotient of	Dic₁₅₂₉₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20388, 6);

Dic5097 #35205128