C₃×#1662 #72377

order	384
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#1662

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₃
prime radical	C₃×#562	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,26)(19,27,29,32)(20,25,21,24)(23,30,31,28)(33,34,35);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,27)(26,31);(4,14)(7,12)(9,16)(10,13)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10683033, D₃×#1662, #10776365, …
maximal subgroup of	C₃×#7525, C₂×C₃×#1662, C₃×#18987, …
maximal normal subgroup of	C₃×#7525, C₂×C₃×#1662, C₃×#18987, …
maximal quotient of	C₃×#7762, C₃×#7769, C₃×#7773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4953);