C₃×#13309 #10631916

order	768
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13309

Subgroups

	subgroup	G/H
center	C₃×C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1999
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 96

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6297);