C₃⋊#168 #1193

order	120
isoclinism	isoclinic class of D₁₅
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₅) (order 480) · Inn(G) = D₁₅ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₅
commutator, hypocenter	C₃×C₅	C₈
Fitting, cosocle	C₃×C₄×C₅	C₂
socle, prime radical	C₂×C₃×C₅	C₄
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₄

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 3);