order	468
isoclinism	isoclinic class of C₂²⋊(C₁₃⋊C₃)
automorphism	?
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₃×C₂²⋊(C₁₃⋊C₃)
unambiguous semidirect product	(C₂²×C₃)⋊(C₁₃⋊C₃)
unambiguous semidirect product	(C₃×C₁₃)⋊A₄

Splittings

	subgroup	G/H
center, hypercenter	C₃	C₂²⋊(C₁₃⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₁₃	C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₃	C₃
LCS	G ▷ C₂²×C₁₃
UCS	C₁ ◁ C₃

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 156

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 49);