C₅⋊#226 #6744

order	240
isoclinism	isoclinic class of C₅⋊#226
automorphism	Aut(G) = C₂×AGL(1,𝔽₅)×S₄ (order 960) · Inn(G) = C₅⋊S₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

C₅ ⋊ #226

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₅⋊S₄
commutator, prime radical, cosocle, hypocenter	C₅×SL(2,𝔽₃)	C₂
Fitting	C₅×Q₈	D₃
socle	C₂×C₅	S₄
LCS	G ▷ C₅×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

26 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 105);