C₃×#13670 #10632277

order	768
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13670

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2184
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#131 ◁ C₃×#504 ◁ G

278 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3)(5,12)(6,17)(7,21)(8,10)(9,23)(13,35)(14,40)(15,41)(16,34)(18,45)(19,49)(20,27)(22,29)(24,52)(25,51)(26,53)(28,30)(31,33)(32,36)(37,46)(38,60)(39,59)(42,56)(43,47)(48,66)(50,61)(54,58)(55,65)(62,64)(63,67);(4,5)(6,13)(7,22)(8,14)(9,29)(10,31)(11,16)(12,21)(15,23)(17,36)(18,46)(19,50)(20,39)(24,41)(25,56)(26,54)(27,38)(28,42)(30,58)(32,47)(33,51)(34,53)(35,45)(37,44)(40,52)(43,61)(48,67)(49,62)(55,64)(57,63)(59,66)(60,65);(4,6,11,20)(5,13,16,39)(7,18,26,48)(8,27,10,17)(9,19,30,49)(12,32,34,60)(14,38,31,36)(15,37,42,63)(21,47,53,65)(22,46,54,67)(23,44,28,57)(24,55,25,43)(29,50,58,62)(33,59,40,35)(41,64,56,61)(45,51,66,52)

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6658);