C₃×#10664 #10629271

order	768
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#10664

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#399
prime radical	C₂³×C₃	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 304 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)(4,5,8,12,9,13,7,11)(6,14,15,19,16,18,17,10)(20,21,23,27,24,28,32,25)(22,29,30,34,31,35,33,26);(4,6,7,15,9,16,8,17)(5,10,11,14,13,19,12,18)(21,25)(23,32)(26,29)(27,28)(30,33)(34,35);(4,7,9,8)(5,11,13,12)(6,15,16,17)(10,18,19,14)(20,22,24,31)(21,26,28,34)(23,33,32,30)(25,29,27,35)

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, 3652);