A₄×D₉ #6260

order	216
isoclinism	isoclinic class of A₄×D₉
automorphism	Aut(G) = S₄×AGL(1,ℤ/9) (order 1296) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

A₄×D₉

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

40 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 97);