C₅²×A₄ #64499

order	300
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₅²×A₄
unambiguous semidirect product	C₂²⋊(C₃×C₅²)
unambiguous semidirect product	(C₂²×C₅)⋊(C₃×C₅)

Fitting, socle, Hall	C₂²×C₅² ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₅²

	subgroup	G/H
center, hypercenter	C₅²	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₅²	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₅²

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 42);