C₂×H(3,𝔽₃)×AGL(1,ℤ/8) #13154702

order	1728
nilpotency class	3
isoclinism	isoclinic class of H(3,𝔽₃)×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃²

Aliases

direct product	C₂×H(3,𝔽₃)×AGL(1,ℤ/8)
unambiguous semidirect product	(C₂×C₃²×AGL(1,ℤ/8))⋊C₃

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

1639 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 19 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20765);