C₂²×D₁₅ #1236

order	120
isoclinism	isoclinic class of D₁₅
automorphism	Aut(G) = D₃×AGL(1,𝔽₅)×S₄ (order 2880) · Inn(G) = D₁₅ · Out(G) = C₄×S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×D₁₅

	subgroup	G/H
center, hypercenter	C₂²	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂²

64 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(120, 46);