C₃⋊#3689 #88609

order	432
isoclinism	isoclinic class of C₃⋊#3688
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, hypercenter	C₂	C₃⋊((C₂²⋊C₉)⋊C₂)
commutator, cosocle, hypocenter	C₃×Q₈⋊C₉	C₂
Frattini	C₂×C₃	C₃⋊S₄
Fitting	C₃²×Q₈	D₃
socle	C₂×C₃²	S₄
LCS	G ▷ C₃×Q₈⋊C₉
UCS	C₁ ◁ C₂

82 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,8,3,4,7,6,5)(10,11)(13,15)(14,18)(16,17)(19,20);(9,10,11);(2,5,8)(3,6,7)(12,20,15,18,17,16,14,13,19)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 256);