D₃×#731 #92275

order	486
automorphism	Aut(G) = C₃×D₃×((C₃⋊D₃)⋊C₃)⋊C₂ (order 1944) · Inn(G) = D₃×H(3,𝔽₃) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	D₃×#731
unambiguous semidirect product	C₃⋊(C₂×#731)

Splittings

Fitting, Sylow

C₃×#731 ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	D₃×H(3,𝔽₃)
commutator, cosocle	C₃³	C₂×C₃²
Frattini	C₃²	C₃²×D₃
Fitting	C₃×#731	C₂
socle	C₃²	C₂×H(3,𝔽₃)
prime radical	D₃×H(3,𝔽₃)	C₃
hypercenter	#731	D₃
hypocenter	C₃	C₂×#731
LCS	G ▷ C₃³ ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #731

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#4176, D₃×#4177, D₃×#4184, …
maximal normal subgroup of	D₃×#4176, D₃×#4177, D₃×#4184, …
maximal quotient of	Dic₃×#731, C₂×D₃×#731, D₃×#6857, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 120);

D3×#731 #92275

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

D₃×#731 #92275