C₂×#543⋊C₃ #85649

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product	C₂×#543⋊C₃
unambiguous semidirect product	(C₂×#543)⋊C₃

Fitting, Sylow	C₂×#543 ⋊ C₃
derived	#543 ⋊ C₂×C₃

	subgroup	G/H
center, socle, hypercenter	C₂³	C₂²⋊A₄
commutator, cosocle, hypocenter	#543	C₂×C₃
Frattini	C₂²	C₂×C₂²⋊A₄
Fitting	C₂×#543	C₃
LCS	G ▷ #543
UCS	C₁ ◁ C₂³

306 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18225);