C₂×D₄×H(3,𝔽₃) #88757

order	432
nilpotency class	2
isoclinism	isoclinic class of D₄×H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃²

Aliases

direct product	C₂×D₄×H(3,𝔽₃)
unambiguous semidirect product	C₄⋊(C₂²×H(3,𝔽₃))
unambiguous semidirect product	(C₂×C₃²×D₄)⋊C₃

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂³×C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

297 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 404);