#6864

order	243
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	Aut(G) = #72277433 (order 4374) · Inn(G) = C₃² · Out(G) = C₃×#4183
homology	Abelianization (H₁) = C₉² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Subgroups

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

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 2 generators on 36 points:

(10,11,14,12,16,22,18,25,30,13,17,21,19,26,29,27,33,35,20,15,23,28,24,31,34,32,36);(1,2,4,3,5,7,6,8,9)(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36)

Polycyclic presentation:

3^5|a^3=c,b^3=d,c^3=e,b^a=be

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#14546335, C₂×#14546335, C₂×#72221799
Fitting subgroup of	#6864⋊C₂
maximal subgroup of	#6864⋊C₂, C₂×#6864, #10624462, …
maximal normal subgroup of	#6864⋊C₂, C₂×#6864, #10624462, …
maximal quotient of	C₂×#6864, #10624443, #10624463, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 11);