D₉⋊SL(2,𝔽₃) #88616

order	432
isoclinism	isoclinic class of #88614
automorphism	Aut(G) = C₂×A₄×AGL(1,ℤ/9) (order 1296) · Inn(G) = D₉⋊A₄ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

Q₈×C₉ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₂	D₉⋊A₄
commutator, Fitting, cosocle, hypocenter	Q₈×C₉	C₂×C₃
Frattini, socle	C₂×C₃	D₃×A₄
LCS	G ▷ Q₈×C₉
UCS	C₁ ◁ C₂

57 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:

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

Polycyclic presentation:

2,3,2^2,3,2,3|c^2=f,d^2=f,e^3=g^2,e^a=e^2g,g^a=g^2,c^b=cd,d^b=c,e^b=eg,d^c=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 263);