C₃×#13199 #10631806

order	768
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#13199

Subgroups

	subgroup	G/H
center	C₃×C₈	#124
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1955
prime radical	C₃×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ G

406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 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)(4,5,9,15,10,16,31,45,11,17,32,46,8,14,19,34)(6,18,23,47,21,36,48,44,22,42,30,58,28,12,33,37)(7,13,26,40,25,39,54,63,27,41,57,66,29,43,49,60)(20,50,53,65,24,35,55,62,52,38,59,64,51,61,56,67);(4,6,19,48,8,28,32,23,11,22,31,33,10,21,9,30)(5,12,34,47,14,42,46,37,17,36,45,58,16,18,15,44)(7,24,54,53,29,20,26,56,27,51,49,59,25,52,57,55)(13,38,63,62,43,35,40,65,41,50,60,67,39,61,66,64);(4,7)(5,13)(6,20,22,52)(8,29)(9,26)(10,25)(11,27)(12,35,36,61)(14,43)(15,40)(16,39)(17,41)(18,38,42,50)(19,49)(21,24,28,51)(23,53,30,59)(31,54)(32,57)(33,56,48,55)(34,60)(37,62,44,67)(45,63)(46,66)(47,64,58,65)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6187);