C₃×#10222 #10628829

order	768
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10222

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

484 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 114 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)(4,5,9,13)(6,14,17,18)(7,10,8,12)(11,15,19,16)(22,29)(23,24)(26,30)(27,33)(31,32)(34,35);(5,10)(7,8)(11,18)(12,13)(14,19)(15,16)(20,21)(22,30)(23,27)(24,33)(25,28)(26,32)(29,34)(31,35);(4,6,9,17)(5,11,13,19)(7,16,8,15)(10,18,12,14)(20,22)(21,26)(23,32)(24,29)(25,31)(27,35)(28,34)(30,33)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3210);