ΓL(1,𝔽₁₆) #306

order	60
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = D₃×AGL(1,𝔽₅) (order 120) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72224613
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

general semilinear	ΓL(1,𝔽₁₆)
unambiguous semidirect product	C₃⋊AGL(1,𝔽₅)

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₅	C₄
prime radical, cosocle	C₃×D₅	C₂
LCS	G ▷ C₃×C₅
UCS	C₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 15

A minimal faithful permutation representation — 2 generators on 8 points:

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

Polycyclic presentation:

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

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(60, 7);