C₃×#10210 #10628817

order	768
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10210

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

440 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 192

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

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3198);