C₃²×#6900 #14555293

order	2187
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃⁵ · Schur multiplier (H₂) = C₃⁸

Aliases

	subgroup	G/H
center, socle	C₃⁴	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃² ▷ C₁

6740 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 729

A minimal faithful permutation representation — 5 generators on 24 points:

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

459 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 9152);