order	432
isoclinism	isoclinic class of D₃×(C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = D₃×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 1296) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

direct product	D₃×(C₂²⋊C₉)⋊C₂
unambiguous semidirect product	(C₃×(C₂²⋊C₉)⋊C₂)⋊C₂
unambiguous semidirect product	(C₃⋊((C₂²⋊C₉)⋊C₂))⋊C₂
unambiguous semidirect product	(D₃×C₂²⋊C₉)⋊C₂

Splittings

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

123 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 123 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 522);