3₋¹⁺²×C₃²⋊C₉ #14552058

order	2187
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)²
automorphism	?
homology	Abelianization (H₁) = C₃³×C₉ · Schur multiplier (H₂) = C₃⁶

Aliases

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃²	C₃³×C₉
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

2156 conjugacy classes of subgroups; only the 638 normal subgroups are shown.

normal subgroups only

This group has 2156 conjugacy classes of subgroups — too many to draw. Only its 638 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 243

A minimal faithful permutation representation — 4 generators on 27 points:

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 5917);