C₃×#10045 #10628652

order	768
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10045

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

444 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 51
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3)(4,5)(6,12)(7,15)(8,11)(9,14)(10,17)(13,19)(16,18)(20,21,26,31)(22,32,35,43)(23,28,39,46)(24,30,25,29)(27,36,42,34)(33,48,49,44)(37,51,50,41)(38,47,40,45);(9,18)(12,19)(20,22,26,35)(21,27,31,42)(23,37,39,50)(24,36,25,34)(28,44,46,48)(29,43,30,32)(33,40,49,38)(41,47,51,45);(4,6)(5,9)(7,16)(8,13)(10,19)(11,18)(12,15)(14,17)(20,23,26,39)(21,28,31,46)(22,33,35,49)(24,38,25,40)(27,41,42,51)(29,45,30,47)(32,44,43,48)(34,37,36,50)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3033);