C₃×#10389 #10628996

order	768
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#10389

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

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

normal subgroups only

This group has 406 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)(4,5,9,12)(6,13)(7,11,8,17)(10,14)(15,19)(16,18)(20,21)(22,25)(23,26)(24,27)(28,33)(29,32)(30,34)(31,35);(6,14,15,16)(10,13,18,19)(20,22)(21,25)(23,28)(24,29)(26,33)(27,32)(30,35)(31,34);(4,6)(5,10)(7,14)(8,16)(9,15)(11,19)(12,18)(13,17)(20,23,24,30)(21,26,27,34)(22,28,29,35)(25,31,32,33)

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, 3377);