C₃×#41 #245

order	48
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×S₄ (order 96) · Inn(G) = C₂² · Out(G) = C₂²×D₃
holomorph	G ⋊ Aut(G) = D₃×#11715828
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center	C₃×C₄	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₃
socle	C₂×C₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 11 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 47);