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H(3,𝔽₃)⋊C₂₇ #10624447

order	729
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₇ · Schur multiplier (H₂) = C₃²

Aliases

unambiguous semidirect product

H(3,𝔽₃)⋊C₂₇

Subgroups

	subgroup	G/H
center	C₃×C₉	H(3,𝔽₃)
commutator	C₃²	C₃×C₂₇
Frattini, cosocle	C₃²×C₉	C₃²
socle	C₃²	C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃² ▷ C₁

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 243

Polycyclic presentation:

3^6|a^3=d,d^3=f,b^a=bc,c^a=ce,c^b=ce

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13090445
Fitting subgroup of	#272⋊C₂₇, #13090445, #13090446
maximal subgroup of	C₂×H(3,𝔽₃)⋊C₂₇, #272⋊C₂₇, #13090445, …
maximal normal subgroup of	C₂×H(3,𝔽₃)⋊C₂₇, #272⋊C₂₇, #13090445, …
maximal quotient of	C₂×H(3,𝔽₃)⋊C₂₇, #14546143, #14546144, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 7);