A₄×#170 #91890

order	480
isoclinism	isoclinic class of A₄×AGL(1,𝔽₅)
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = A₄×AGL(1,𝔽₅) · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×#170
unambiguous semidirect product	(C₂²×#170)⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄×AGL(1,𝔽₅)
commutator, hypocenter	C₂²×C₅	C₃×C₈
Fitting, socle	C₂³×C₅	C₃×C₄
prime radical	C₂×C₅×A₄	C₄
cosocle	C₂²×Dic₅	C₂×C₃
LCS	G ▷ C₂²×C₅
UCS	C₁ ◁ C₂

52 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 17 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 966);