H(3,𝔽₃)×#123 #11721905

order	864
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃²

Aliases

direct product	H(3,𝔽₃)×#123
unambiguous semidirect product	(C₃²×#123)⋊C₃

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²×C₃	C₂³×C₃²
prime radical	C₂×D₄×H(3,𝔽₃)	C₂
LCS	G ▷ C₂²×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

429 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1872);