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H(3,𝔽₃)×C₂₂₃ #20258253

order	6021
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₂₂₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product	H(3,𝔽₃)×C₂₂₃
unambiguous semidirect product	C₃²⋊(C₃×C₂₂₃)

Subgroups

	subgroup	G/H
center, socle	C₃×C₂₂₃	C₃²
commutator, Frattini	C₃	C₃²×C₂₂₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₂₂₃ ◁ G

22 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2453 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#31825587, #31825590, #31825593, …
maximal subgroup of	AΣL(1,𝔽₂₇)×C₂₂₃, #731×C₂₂₃, #732×C₂₂₃, …
maximal normal subgroup of	AΣL(1,𝔽₂₇)×C₂₂₃, #731×C₂₂₃, #732×C₂₂₃, …
maximal quotient of	C₃²⋊C₉×C₂₂₃, AΣL(1,𝔽₂₇)×C₂₂₃, #731×C₂₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6021, 3);