C₃×#10115 #10628722

order	768
nilpotency class	3
isoclinism	isoclinic class of #10104
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10115

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

466 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

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, 3103);