C₂×S₅ #6828

order	240
isoclinism	isoclinic class of S₅
automorphism	Aut(G) = C₂×S₅ (order 240) · Inn(G) = S₅ · Out(G) = C₂
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

general orthogonal	GO(3,𝔽₅)
direct product	C₂×S₅
unambiguous semidirect product	A₅⋊C₂²
unambiguous semidirect product	(C₂×A₅)⋊C₂

socle	C₂×A₅ ⋊ C₂
derived	A₅ ⋊ C₂²

	subgroup	G/H
center, Fitting, hypercenter	C₂	S₅
commutator, cosocle, hypocenter	A₅	C₂²
Frattini	C₁	G
socle	C₂×A₅	C₂
LCS	G ▷ A₅
UCS	C₁ ◁ C₂

57 conjugacy classes of subgroups.

minimal degree	μ(G) = 7
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 2 generators on 7 points:

(6,7);(1,2)(3,4,5,6,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 189);