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D₉×H(3,𝔽₃) #92254

order	486
isoclinism	isoclinic class of D₉×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃²

Aliases

D₉×H(3,𝔽₃)

Splittings

C₉×H(3,𝔽₃) ⋊ C₂

Subgroups

	subgroup	G/H
center	C₃	C₃²×D₉
commutator, cosocle	C₃×C₉	C₂×C₃²
Frattini, socle	C₃²	C₃²×D₃
Fitting	C₉×H(3,𝔽₃)	C₂
hypercenter	H(3,𝔽₃)	D₉
hypocenter	C₉	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₃ ◁ H(3,𝔽₃)

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₉×(C₃⋊D₃)⋊C₃, D₉×#272, C₂×D₉×H(3,𝔽₃), …
maximal normal subgroup of	D₉×(C₃⋊D₃)⋊C₃, D₉×#272, C₂×D₉×H(3,𝔽₃), …
maximal quotient of	H(3,𝔽₃)×Dic₉, C₂×D₉×H(3,𝔽₃), D₉×C₃²⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 99);