C₃×#1696 #72411

order	384
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#1696

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ G

140 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 384

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

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

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10709105, #10709112, #10709273, …
maximal subgroup of	C₃×#12388, C₃×#12389, C₃×#12412, …
maximal normal subgroup of	C₃×#12388, C₃×#12389, C₃×#12412, …
maximal quotient of	C₃×#8653, C₃×#9224, C₃×#9576, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4987);