Dic₁₅ #302

order	60
isoclinism	isoclinic class of D₁₅
automorphism	Aut(G) = C₂×D₃×AGL(1,𝔽₅) (order 240) · Inn(G) = D₁₅ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = #72224650
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₁₅
unambiguous semidirect product	C₃⋊Dic₅

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

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 12 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(60, 3);