C₃×#9937 #10628544

order	768
nilpotency class	3
isoclinism	isoclinic class of #9930
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9937

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

526 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2925);