C₃×#12295 #10630902

order	768
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12295

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₃×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#424 ◁ G

290 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 74 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)(6,9,13,16)(7,10,14,17)(12,18,19,15)(20,21,26,32)(22,33,36,48)(23,27,40,44)(24,31,25,30)(28,41,46,35)(29,39,43,37)(34,47,38,45)(42,50,49,51);(4,6)(5,9)(7,12)(8,13)(10,15)(11,16)(14,19)(17,18)(21,27)(23,37)(24,25)(28,42)(29,31)(30,43)(32,44)(33,47)(34,50)(35,41)(38,51)(39,40)(45,48)(46,49);(4,7,8,14)(5,10,11,17)(6,12,13,19)(9,15,16,18)(20,22)(21,28)(23,38)(24,41)(25,35)(26,36)(27,42)(29,47)(30,48)(31,33)(32,46)(34,40)(37,51)(39,50)(43,45)(44,49)

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, 5283);