C₅²×C₄²⋊C₃ #11925173

order	1200
isoclinism	isoclinic class of C₄²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₅² · Schur multiplier (H₂) = C₄×C₅

Aliases

direct product	C₅²×C₄²⋊C₃
unambiguous semidirect product	C₄²⋊(C₃×C₅²)
unambiguous semidirect product	(C₄²×C₅)⋊(C₃×C₅)

Fitting, Hall	C₄²×C₅² ⋊ C₃
derived, Sylow	C₄² ⋊ C₃×C₅²

	subgroup	G/H
center, hypercenter	C₅²	C₄²⋊C₃
commutator, cosocle, hypocenter	C₄²	C₃×C₅²
Frattini	C₂²	C₅²×A₄
Fitting	C₄²×C₅²	C₃
socle	C₂²×C₅²	A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₅²

80 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 300

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3,4,5)(11,12,15)(13,17,22)(14,18,21)(16,20,19);(6,7,8,9,10)(11,13,14,19)(12,18)(15,22,21,20)(16,17)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 383);