C₃⋊((C₂²⋊C₉)⋊C₂) #6257

order	216
isoclinism	isoclinic class of C₃⋊((C₂²⋊C₉)⋊C₂)
automorphism	Aut(G) = #72276271 (order 11664) · Inn(G) = G · Out(G) = #272
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂×C₃

Aliases

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₂²⋊C₉	C₂
Frattini	C₃	C₃⋊S₄
Fitting, socle	C₂²×C₃²	D₃
LCS	G ▷ C₃×C₂²⋊C₉
UCS	C₁

58 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2,3^3,2^2|b^3=d^2,b^a=b^2d,c^a=c^2,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, 94);