(C₂×D₃)⋊AGL(1,𝔽₅) #6735

order	240
isoclinism	isoclinic class of (C₂×D₃)⋊AGL(1,𝔽₅)
automorphism	Aut(G) = C₂²×D₃×AGL(1,𝔽₅) (order 480) · Inn(G) = D₃×AGL(1,𝔽₅) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product	(C₂×D₃)⋊AGL(1,𝔽₅)
unambiguous semidirect product	(C₂×AGL(1,𝔽₅))⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×AGL(1,𝔽₅)
commutator, Fitting, socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂×D₃×D₅	C₂
cosocle	C₂×C₃×D₅	C₂²
hypocenter	C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂

68 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 96);