C₃⋊#3771 #88919

order	432
isoclinism	isoclinic class of C₃⋊AGL(1,𝔽₉)
automorphism	Aut(G) = D₃×AGL(1,𝔽₉)⋊C₄ (order 1728) · Inn(G) = C₃⋊AGL(1,𝔽₉) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

C₃³ ⋊ C₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊AGL(1,𝔽₉)
commutator, hypocenter	C₃³	C₁₆
Fitting, socle, prime radical	C₂×C₃³	C₈
cosocle	C₃×#622	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

30 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 28 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,19)(21,23,24,26,25,27,28,22);(17,18,19)(20,28,24)(21,27,22)(23,25,26)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 566);