C₂×C₃×(C₂²⋊C₉)⋊C₂ #88887

order	432
isoclinism	isoclinic class of (C₂²⋊C₉)⋊C₂
automorphism	Aut(G) = C₂²×((C₂²⋊C₉)⋊C₂)⋊C₃ (order 864) · Inn(G) = (C₂²⋊C₉)⋊C₂ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₂²⋊C₉)⋊C₂
commutator, cosocle, hypocenter	C₂²⋊C₉	C₂²×C₃
Frattini	C₃	C₂×C₃×S₄
Fitting, socle	C₂³×C₃²	D₃
LCS	G ▷ C₂²⋊C₉
UCS	C₁ ◁ C₂×C₃

124 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

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

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 2 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 534);