C₃×#10023 #10628630

order	768
nilpotency class	3
isoclinism	isoclinic class of #10018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10023

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 518 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)(4,5,9,13)(6,14,16,18)(7,10,8,12)(11,17,19,15)(20,21,25,23)(22,28,31,35)(24,32,34,27)(26,33,29,30);(5,10)(7,8)(11,14)(12,13)(15,17)(18,19)(20,22)(21,26)(23,29)(24,33)(25,31)(27,35)(28,32)(30,34);(4,6,9,16)(5,11,13,19)(7,15,8,17)(10,18,12,14)(20,23,25,21)(22,29,31,26)(24,27,34,32)(28,33,35,30)

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, 3011);