C₃⋊C₁₆ #199

order	48
isoclinism	isoclinic class of D₃
automorphism	Aut(G) = C₂×C₄×D₃ (order 48) · Inn(G) = D₃ · Out(G) = C₂×C₄
holomorph	G ⋊ Aut(G) = #72213745
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

C₃⋊C₁₆

C₃ ⋊ C₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₃
commutator, hypocenter	C₃	C₁₆
Fitting, cosocle	C₃×C₈	C₂
socle, prime radical	C₂×C₃	C₈
LCS	G ▷ C₃
UCS	C₁ ◁ C₈

10 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 19 points:

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

Polycyclic presentation:

2^4,3|a^2=b,b^2=c,c^2=d,e^a=e^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 1);