A₄⋊Dic₅ #6746

order	240
isoclinism	isoclinic class of C₅⋊S₄
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = C₅⋊S₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂

Aliases

A₄⋊Dic₅

derived	C₅×A₄ ⋊ C₄
Sylow	C₅ ⋊ SL(2,ℤ/4)

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊S₄
commutator, hypocenter	C₅×A₄	C₄
Fitting, socle	C₂³×C₅	D₃
prime radical, cosocle	C₂×C₅×A₄	C₂
LCS	G ▷ C₅×A₄
UCS	C₁ ◁ C₂

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 107);