H(3,𝔽₃)×#518 #13154537

order	1728
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₄ · Schur multiplier (H₂) = C₂⁵×C₃²

Aliases

direct product	H(3,𝔽₃)×#518
unambiguous semidirect product	(C₃²×#518)⋊C₃

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²×C₃	C₂⁴×C₃²
commutator	C₂×C₃	C₂³×C₃²×C₄
prime radical	C₂²×D₄×H(3,𝔽₃)	C₂
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

1859 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20600);