#6909

order	243
nilpotency class	3
isoclinism	isoclinic class of #6909
automorphism	Aut(G) order 8748
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃³

Aliases

No aliases for this group.

Subgroups

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

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 27

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,5)(3,9,14)(4,8,15)(6,16,12)(7,19,13)(10,21,24)(11,18,25)(17,27,23)(20,26,22);(1,3,10)(2,6,17)(4,12,22)(5,14,24)(7,9,20)(8,18,21)(11,23,13)(15,16,26)(19,25,27);(3,11,12)(6,9,18)(10,22,23)(14,25,16)(17,21,20)(24,26,27)

Polycyclic presentation:

3^5|b^a=bd,d^a=de,c^b=ce

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#92281, #4175⋊C₃, #4183⋊C₃, …
maximal subgroup of	#92281, #4175⋊C₃, #4183⋊C₃, …
maximal normal subgroup of	#92281, #4175⋊C₃, #4183⋊C₃, …
maximal quotient of	C₂×#6909, #10624590, #10624591, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 56);