Finite Group Databasebeta

← Prev (#73276) Next (#73278) →

A₄×D₁₆ #73277

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	A₄×D₁₆
unambiguous semidirect product	(C₂²×D₁₆)⋊C₃

Splittings

C₂²×D₁₆ ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	A₄×D₈
commutator, cosocle	C₂²×C₈	C₂²×C₃
Frattini	C₈	C₂²×A₄
Fitting	C₂²×D₁₆	C₃
socle	C₂³	C₃×D₈
hypercenter	D₁₆	A₄
hypocenter	C₂²	C₃×D₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 20 points:

(2,3,4)(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18);(1,4)(2,3)(5,6)(7,10)(8,17)(9,13)(11,14)(12,15)(16,19)(18,20)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×D₃₂, A₄×SD₆₄, C₂×A₄×D₁₆, …
maximal normal subgroup of	A₄×D₃₂, A₄×SD₆₄, C₂×A₄×D₁₆, …
maximal quotient of	A₄×#357, A₄×#366, #33925⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5853);