C₃×#10648 #10629255

order	768
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#10648

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

236 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 51
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3)(4,5,8,11,9,13,7,12)(6,10,14,19,16,17,15,18)(20,21)(22,30,33,43)(23,27)(24,28)(25,29)(26,32,40,37)(31,46,34,45)(35,41)(36,42)(38,44)(39,48,50,47)(49,51);(4,6)(5,10)(7,15)(8,14)(9,16)(11,19)(12,18)(13,17)(20,22,25,33)(21,26,29,40)(23,34,36,31)(24,37,38,32)(27,39,42,50)(28,43,44,30)(35,47,49,48)(41,46,51,45);(5,11)(6,14)(7,8)(10,17)(12,13)(15,16)(20,23,24,35,25,36,38,49)(21,27,28,41,29,42,44,51)(22,31,32,47,33,34,37,48)(26,39,30,45,40,50,43,46)

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, 3636);