#6858

order	243
nilpotency class	3
isoclinism	isoclinic class of #6856
automorphism	Aut(G) = C₃³×#272 (order 1458) · Inn(G) = H(3,𝔽₃) · Out(G) = C₂×C₃³
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

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

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 81

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

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

Polycyclic presentation:

3^5|a^3=d,b^3=d,b^a=bc,c^a=cd,c^b=ce

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#6858⋊C₂, #6858⋊C₄, C₂×#6858⋊C₂, …
Fitting subgroup of	#6858⋊C₂
maximal subgroup of	#6858⋊C₂, C₂×#6858, C₃×#6858, …
maximal normal subgroup of	#6858⋊C₂, C₂×#6858, C₃×#6858, …
maximal quotient of	C₂×#6858, #10624450, #10624451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 5);