Dic₃⋊#168 #90984

order	480
isoclinism	isoclinic class of #1201
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₈
Fitting, cosocle	C₂×C₃×C₄×C₅	C₂²
socle, prime radical	C₂²×C₃×C₅	C₂×C₄
hypocenter	C₃×C₅	C₄⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2,4,5,3)(7,8)(9,10,11,12);(2,3)(4,5)(6,7,8)(10,12)(13,14,15,17,16,18,19,20)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 60);