(C₂×D₃)⋊Dic₅ #6666

order	240
isoclinism	isoclinic class of #1201
automorphism	Aut(G) = C₂⁴×D₃×AGL(1,𝔽₅) (order 1920) · Inn(G) = D₃×D₅ · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

(C₂×D₃)⋊Dic₅

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

68 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 27);