A₄⋊#170 #91887

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) = C₈ · Schur multiplier (H₂) = C₂

Aliases

derived	C₅×A₄ ⋊ C₈
Sylow	C₅ ⋊ A₄⋊C₈

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅×A₄	C₈
Fitting, socle	C₂³×C₅	Dic₃
prime radical	C₂×C₅×A₄	C₄
cosocle	A₄×Dic₅	C₂
LCS	G ▷ C₅×A₄
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:

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

Polycyclic presentation:

2^3,3,5,2^2|a^2=b,b^2=c,d^a=d^2,e^a=e^3,f^a=g,g^a=f,e^b=e^4,f^d=g,g^d=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 963);