D₉⋊A₄ #6259

order	216
isoclinism	isoclinic class of D₉⋊A₄
automorphism	Aut(G) = A₄×AGL(1,ℤ/9) (order 648) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₉⋊A₄
unambiguous semidirect product	(C₉⋊A₄)⋊C₂

C₂²×C₉ ⋊ C₂×C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂²×C₉	C₂×C₃
Frattini	C₃	D₃×A₄
socle	C₂²×C₃	C₃×D₃
LCS	G ▷ C₂²×C₉
UCS	C₁

38 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 96);