C₃×#13669 #10632276

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

Aliases

direct product

C₃×#13669

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

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

normal subgroups only

This group has 288 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)(4,5)(6,19)(7,24)(8,17)(9,33)(10,15)(11,18)(12,20)(13,21)(14,28)(16,26)(22,53)(23,46)(25,48)(27,36)(29,55)(30,56)(31,41)(32,49)(34,42)(35,47)(37,58)(38,57)(39,52)(40,54)(43,60)(44,61)(45,62)(50,51)(59,67)(63,65)(64,66);(4,6)(5,12)(7,25)(8,21)(9,34)(10,31)(11,23)(13,36)(14,42)(15,37)(16,48)(17,50)(18,39)(19,24)(20,28)(22,26)(27,57)(29,63)(30,59)(32,54)(33,38)(35,64)(40,44)(41,55)(43,53)(45,66)(46,56)(47,65)(49,67)(51,61)(52,62)(58,60);(4,6)(5,13)(7,20)(8,31)(9,22)(10,21)(11,23)(12,36)(14,38)(15,46)(16,40)(17,19)(18,41)(24,50)(25,28)(26,34)(27,58)(29,51)(30,52)(32,64)(33,42)(35,54)(37,56)(39,55)(43,67)(44,48)(45,65)(47,66)(49,53)(57,60)(59,62)(61,63)

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, 6657);