C₂³×#272 #88914

order	432
isoclinism	isoclinic class of #272
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

Fitting	C₂³×H(3,𝔽₃) ⋊ C₂
derived, Sylow	H(3,𝔽₃) ⋊ C₂⁴

	subgroup	G/H
center, socle, hypercenter	C₂³×C₃	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂⁴
Frattini	C₃	C₂³×C₃⋊D₃
Fitting	C₂³×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂³×C₃

737 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 561);