C₂×#560⋊C₃ #85622

order	384
isoclinism	isoclinic class of #560⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₂×#560⋊C₃
unambiguous semidirect product	(#126⋊C₃)⋊C₂²
unambiguous semidirect product	(C₂×#560)⋊C₃

Fitting, Sylow	C₂×#560 ⋊ C₃
derived	C₄² ⋊ C₂³×C₃

	subgroup	G/H
center, hypercenter	C₂	#560⋊C₃
commutator, cosocle, hypocenter	C₄²	C₂³×C₃
Frattini	C₂²	C₂³×A₄
Fitting	C₂×#560	C₃
socle	C₂³	C₂²×A₄
LCS	G ▷ C₄²
UCS	C₁ ◁ C₂

304 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 24

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18198);