D₃×#157 #6319

order	216
isoclinism	isoclinic class of D₃×#157
automorphism	Aut(G) = D₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

C₃³ ⋊ C₂×C₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃³	C₂×C₄
prime radical	D₃×C₃⋊D₃	C₂
cosocle	C₃×C₃⋊D₃	C₂²
LCS	G ▷ C₃³
UCS	C₁

60 conjugacy classes of subgroups.

minimal degree	μ(G) = 9
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 9 points:

(1,2,3)(4,5)(6,8,9,7);(2,3)(4,6,9)(5,8,7)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 156);