C₃×#10211 #10628818

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

Aliases

direct product

C₃×#10211

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

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

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, 3199);