C₃×#13544 #10632151

order	768
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#13544

Subgroups

	subgroup	G/H
center	C₃×C₄	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#2101
prime radical	C₂³×C₃	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#405 ◁ G

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

normal subgroups only

This group has 212 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) = 67
transitive degree	μ_t(G) = 192

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

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

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, 6532);