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#10624443

order	729
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₉×C₂₇ · Schur multiplier (H₂) = C₉

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₉²	C₃²
commutator	C₃	C₉×C₂₇
socle, prime radical	C₃²	C₉²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉² ◁ G
LECS	G ▷ C₉² ▷ C₃² ▷ C₁

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 729

Polycyclic presentation:

3^6|a^3=c,b^3=d,c^3=e,d^3=f,b^a=be

Conjugacy classes & character table

297 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72277430, #72277445
Fitting subgroup of	#10624443⋊C₂
maximal subgroup of	C₂×#10624443, #10624443⋊C₂, #14546223, …
maximal normal subgroup of	C₂×#10624443, #10624443⋊C₂, #14546223, …
maximal quotient of	C₂×#10624443, #14546147, #14546222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 3);