#3689⋊C₃ #88598

order	432
isoclinism	isoclinic class of #3688⋊C₃
automorphism	Aut(G) = C₂×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 432) · Inn(G) = ((C₂²⋊C₉)⋊C₂)⋊C₃ · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

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

54 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 245);