C₃×#10079 #10628686

order	768
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10079

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂×C₃×#496	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 658 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) = 23
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(9,14)(10,15)(11,12)(17,20)(18,22)(21,23);(4,5)(6,7)(8,9)(10,13)(11,15)(12,14)(18,22)(20,23);(5,7)(8,10,13,9)(11,15,12,14)(16,17)(18,23)(19,21)(20,22)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3067);