C₃×#10217 #10628824

order	768
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10217

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

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 51
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3)(4,5)(6,13)(7,9)(8,18)(10,12)(11,14)(15,19)(16,17)(20,21,26,31)(22,32,35,46)(23,36,39,44)(24,30,25,29)(27,40,42,34)(28,37,45,38)(33,41,49,43)(47,50,51,48);(4,6,9,15)(5,11,7,17)(8,16,10,14)(12,19,18,13)(20,22,26,35)(21,27,31,42)(23,33,39,49)(24,40,25,34)(28,43,45,41)(29,46,30,32)(36,47,44,51)(37,50,38,48);(4,7,9,5)(6,13,15,19)(8,18,10,12)(11,14,17,16)(20,23,26,39)(21,28,31,45)(22,33,35,49)(24,38,25,37)(27,41,42,43)(29,36,30,44)(32,47,46,51)(34,50,40,48)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3205);