H(3,𝔽₃)×#492 #13145235

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×#465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃²×C₄

Aliases

direct product	H(3,𝔽₃)×#492
unambiguous semidirect product	(C₃²×#492)⋊C₃

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	H(3,𝔽₃)×#127	C₂
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

781 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 231 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) = 288

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 11298);