C₂×C₃×D₃×A₄ #89116

order	432
isoclinism	isoclinic class of D₃×A₄
automorphism	Aut(G) = C₂×D₃²×S₄ (order 1728) · Inn(G) = D₃×A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product	C₂×C₃×D₃×A₄
unambiguous semidirect product	C₃⋊(C₂²×C₃×A₄)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₃×A₄)
unambiguous semidirect product	(C₂³×D₃)⋊C₃²
unambiguous semidirect product	(C₂³×C₃×D₃)⋊C₃

Fitting, socle	C₂³×C₃² ⋊ C₂×C₃
derived	C₂²×C₃ ⋊ C₂²×C₃²

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃×A₄
commutator, cosocle, hypocenter	C₂²×C₃	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₂×C₃
LCS	G ▷ C₂²×C₃
UCS	C₁ ◁ C₂×C₃

232 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

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

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 763);