C₃⋊#170 #1197

order	120
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₅) (order 240) · Inn(G) = ΓL(1,𝔽₁₆) · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = #72231278
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₈
Fitting, socle, prime radical	C₂×C₃×C₅	C₄
cosocle	C₃×Dic₅	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 7);