Finite Group Databasebeta

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#10624833

order	729
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₂₇ · Schur multiplier (H₂) = C₃²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈₁	C₃²
commutator, socle	C₃	C₃²×C₂₇
Frattini, cosocle	C₂₇	C₃³
prime radical	H(3,𝔽₃)	C₂₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₈₁ ◁ G
LECS	G ▷ C₂₇ ▷ C₉ ▷ C₃ ▷ C₁

77 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 243
transitive degree	μ_t(G) = 243

Polycyclic presentation:

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

Conjugacy classes & character table

297 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13091280, #13091281
maximal subgroup of	C₂×#10624833, #13091280, #13091281, …
maximal normal subgroup of	C₂×#10624833, #13091280, #13091281, …
maximal quotient of	C₂×#10624833, #14551441, #14551446, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 393);