C₉⋊S₄ #6256

order	216
isoclinism	isoclinic class of C₉⋊S₄
automorphism	Aut(G) = #72276182 (order 3888) · Inn(G) = G · Out(G) = C₃×D₃
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃

Aliases

unambiguous semidirect product	C₉⋊S₄
unambiguous semidirect product	A₄⋊D₉

Fitting	C₂²×C₉ ⋊ D₃
derived	C₉×A₄ ⋊ C₂

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉×A₄	C₂
Frattini	C₃	C₃⋊S₄
Fitting	C₂²×C₉	D₃
socle	C₂²×C₃	C₃⋊D₃
LCS	G ▷ C₉×A₄
UCS	C₁

49 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 13 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 93);