Finite Group Databasebeta

← Prev (#92377) Next (#92379) →

C₃×D₃×H(3,𝔽₃) #92378

order	486
isoclinism	isoclinic class of D₃×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product	C₃×D₃×H(3,𝔽₃)
unambiguous semidirect product	C₃⋊(C₂×C₃×H(3,𝔽₃))
unambiguous semidirect product	(C₃²×D₃)⋊C₃²

Splittings

C₃²×H(3,𝔽₃) ⋊ C₂

Subgroups

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

393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 15 points:

(5,6)(8,11,12)(10,14,15);(4,5,6)(7,8,10)(9,12,14)(11,15,13);(1,2,3)

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₃×(C₃⋊D₃)⋊C₃, D₃²×H(3,𝔽₃), D₃×C₃⋊((C₃⋊D₃)⋊C₃), …
maximal normal subgroup of	C₃×D₃×(C₃⋊D₃)⋊C₃, D₃²×H(3,𝔽₃), D₃×C₃⋊((C₃⋊D₃)⋊C₃), …
maximal quotient of	C₃×Dic₃×H(3,𝔽₃), C₂×C₃×D₃×H(3,𝔽₃), C₃×D₃×C₃²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 223);