C₃×#10385 #10628992

order	768
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#10385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

396 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 90 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)(4,5,7,10)(6,12)(8,11,16,18)(9,17)(13,14)(15,19)(20,21)(22,25)(23,30,32,26)(24,27)(28,33,31,35)(29,34);(4,5)(6,13,15,9)(7,10)(8,11)(12,14,19,17)(16,18)(20,22,24,29)(21,25,27,34)(23,31)(26,35)(28,32)(30,33);(4,6)(5,9)(7,15)(8,17)(10,13)(11,19)(12,18)(14,16)(20,23,24,32)(21,26,27,30)(22,28,29,31)(25,33,34,35)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3373);