order	500
isoclinism	isoclinic class of (C₅⋊D₅)⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₂×(C₅⋊D₅)⋊C₅
unambiguous semidirect product	(C₂×C₅⋊D₅)⋊C₅

Splittings

	subgroup	G/H
center, hypercenter	C₂	(C₅⋊D₅)⋊C₅
commutator, cosocle, hypocenter	C₅²	C₂²×C₅
Frattini	C₅	C₂×C₅×D₅
Fitting	C₂×H(3,𝔽₅)	C₂
socle	C₂×C₅	C₅×D₅
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂

43 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 50

A minimal faithful permutation representation — 2 generators on 27 points:

(4,11,13,20,17,6,7,19,21,12)(5,9)(8,25,26,22,23,15,16,24,14,27)(10,18);(1,2)(3,4,8,15,6)(5,7,14,24,12)(9,17,26,22,11)(10,13,16,27,20)(18,21,23,25,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 30);