C₂×#4169 #66371

order	324
isoclinism	isoclinic class of #4169
automorphism	Aut(G) = C₂×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃ (order 1944) · Inn(G) = #4169 · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

	subgroup	G/H
center, hypercenter	C₂	#4169
commutator, cosocle, hypocenter	C₃×C₉	C₂²×C₃
Frattini	C₃²	C₂×C₃×D₃
Fitting	C₂×C₃²⋊C₉	C₂
socle	C₂×C₃²	C₃×D₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂

83 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 20 points:

(3,4,6)(5,7,10,9,8,11)(13,15)(14,18)(16,17)(19,20);(1,2)(4,7,8)(6,10,11)(12,20,15,18,17,16,14,13,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 63);