C₉⋊GL(2,𝔽₃) #88606

order	432
isoclinism	isoclinic class of C₉⋊#226
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

Fitting	Q₈×C₉ ⋊ D₃
derived	C₉×SL(2,𝔽₃) ⋊ C₂

	subgroup	G/H
center, hypercenter	C₂	C₉⋊S₄
commutator, cosocle, hypocenter	C₉×SL(2,𝔽₃)	C₂
Frattini, socle	C₂×C₃	C₃⋊S₄
Fitting	Q₈×C₉	D₃
LCS	G ▷ C₉×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

76 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 17 points:

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

Polycyclic presentation:

2,3^3,2^3|c^3=d^2,e^2=g,f^2=g,b^a=b^2,c^a=c^2d,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, 253);