C₅⋊GL(2,𝔽₃) #6745

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

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

32 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

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

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

Polycyclic presentation:

2,3,5,2^3|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, 106);