C₅²×C₂²⋊A₄ #11925823

order	1200
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₅² · Schur multiplier (H₂) = C₂⁴×C₅

Aliases

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

Fitting, socle, Hall	C₂⁴×C₅² ⋊ C₃
derived, Sylow	C₂⁴ ⋊ C₃×C₅²

	subgroup	G/H
center, hypercenter	C₅²	C₂²⋊A₄
commutator, cosocle, hypocenter	C₂⁴	C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅²	C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₅²

272 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 300

A minimal faithful permutation representation — 3 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1200, 1033);