C₃×#10213 #10628820

order	768
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#10213

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 498 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) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5)(6,13)(7,18)(8,11)(9,12)(10,15)(14,17)(16,19)(20,21)(22,30)(23,28)(24,27)(25,29)(26,31)(32,35)(33,34);(4,6,9,17)(5,10,12,19)(7,16,8,15)(11,14,18,13)(20,22,25,32)(21,26,29,34)(23,33,24,31)(27,35,28,30);(4,5)(6,14)(7,11)(8,18)(9,12)(10,15)(13,17)(16,19)(21,27)(23,24)(26,30)(28,29)(31,33)(34,35)

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, 3201);