3₋¹⁺²×#121 #11721900

order	864
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³

Aliases

direct product	3₋¹⁺²×#121
unambiguous semidirect product	(C₉×#121)⋊C₃

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂³×C₃²
prime radical	C₃²×#121	C₃
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

376 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

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

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2,4,3,6,8,7,5,9)(11,13)(15,17);(2,5,6)(4,8,9)(14,15,16,17);(10,11)(12,13)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1867);