C₃×#13313 #10631920

order	768
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13313

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#131 ◁ G

556 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 39
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(4,5)(6,7)(8,9)(10,21)(11,15)(12,19)(13,18)(14,20)(16,24)(17,26)(22,35)(23,33)(25,34)(27,30)(28,31)(29,39)(32,38)(36,37);(9,15)(10,22)(11,26)(12,13)(16,32)(17,18)(19,30)(20,31)(21,36)(23,38)(24,25)(27,28)(29,34)(33,35);(5,7)(8,10)(9,16)(11,23)(12,22)(13,24)(14,25)(15,29)(17,33)(18,34)(19,21)(20,35)(26,37)(27,39)(28,38)(30,32)(31,36)

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, 6301);