#733

order	81
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	Aut(G) = #92285 (order 486) · Inn(G) = H(3,𝔽₃) · Out(G) = C₂×C₃²
holomorph	G ⋊ Aut(G) = #72222318
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Subgroups

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

13 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

3^4|a^3=d,b^3=d^2,b^a=bc,c^a=cd

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#733⋊C₂, #733⋊C₄, C₂×#733⋊C₂, …
Fitting subgroup of	#733⋊C₂
maximal subgroup of	#733⋊C₂, C₂×#733, #6878, …
maximal normal subgroup of	#733⋊C₂, C₂×#733, #6878, …
maximal quotient of	C₂×#733, #6858, #6860, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(81, 10);