C₅⋊(#41⋊C₃) #6747

order	240
isoclinism	isoclinic class of C₅⋊(#41⋊C₃)
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = D₅×A₄ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	C₅⋊(#41⋊C₃)
unambiguous semidirect product	#713⋊C₃

Hall	#713 ⋊ C₃
Sylow	C₅ ⋊ #41⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×A₄
commutator, Fitting, cosocle, hypocenter	C₅×Q₈	C₂×C₃
socle	C₂×C₅	C₂×A₄
LCS	G ▷ C₅×Q₈
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,5,8)(3,13,12,14,9,7,10,15,4,6,11,16)(18,19)(20,21);(1,11,5,9)(2,15,8,13)(3,4,10,12)(6,7,14,16)(17,18,20,21,19)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 108);