C₅×#226 #6741

order	240
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂×C₄×S₄ (order 192) · Inn(G) = S₄ · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 102);