C₃²×#6891 #14555253

order	2187
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃⁵ · Schur multiplier (H₂) = C₃¹⁰

Aliases

	subgroup	G/H
center, socle	C₃⁴	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃⁶	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃² ▷ C₁

24308 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 243

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

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 9112);