C₃×#10017 #10628624

order	768
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10017

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#392
prime radical	C₂³×C₃	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

528 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 122 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)(4,5,9,12)(6,13,15,19)(7,11,8,16)(10,17,18,14)(20,21)(22,28)(23,31)(24,27)(25,30)(26,33)(29,35)(32,34);(4,6,9,15)(5,10,12,18)(7,14,8,17)(11,19,16,13)(25,34)(28,35);(4,7,9,8)(5,11,12,16)(6,14,15,17)(10,13,18,19)(20,22)(21,25)(23,32)(24,29)(26,35)(27,34)(28,31)(30,33)

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, 3005);