Dic₅₇₃₉ #37525795

order	22956
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇₃₉
unambiguous semidirect product	C₃⋊Dic₁₉₁₃

Splittings

derived, Hall	C₃×C₁₉₁₃ ⋊ C₄
Sylow	C₁₉₁₃ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₉₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₃₉
commutator, hypocenter	C₃×C₁₉₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₉₁₃	C₂
LCS	G ▷ C₃×C₁₉₁₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1920
transitive degree	μ_t(G) = 22956

A minimal faithful permutation representation — 2 generators on 1920 points:

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Polycyclic presentation:

2,3,1913,2|a^2=d,b^a=b^2,c^a=c^1912

Conjugacy classes & character table

5742 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₄₇₈
maximal normal subgroup of	Dic₁₁₄₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22956, 4);

Dic5739 #37525795