A₄⋊AGL(1,𝔽₅) #6831

order	240
isoclinism	isoclinic class of A₄⋊AGL(1,𝔽₅)
automorphism	Aut(G) = AGL(1,𝔽₅)×S₄ (order 480) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅×A₄	C₄
Fitting, socle	C₂²×C₅	Dic₃
prime radical, cosocle	D₅×A₄	C₂
LCS	G ▷ C₅×A₄
UCS	C₁

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,4,3)(6,7,8,9);(2,3,4)(5,6,9,7,8)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 192);