H(3,𝔽₃)×#384 #13144911

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×#120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃²×C₄

Aliases

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

1023 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 455 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) = 576

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10974);