C₃×#12400 #10631007

order	768
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12400

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₃×#3355	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#436 ◁ G

286 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 78 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)(4,5,9,13)(6,14,17,18)(7,12,8,11)(10,16,19,15)(20,21,25,29)(22,30,32,49)(23,34,36,48)(24,39,37,28)(26,41,42,31)(27,44,46,38)(33,50,40,45)(35,47,51,43);(4,6,9,17)(5,10,13,19)(7,16,8,15)(11,18,12,14)(20,22,25,32)(21,26,29,42)(23,35,36,51)(24,40,37,33)(27,45,46,50)(28,47,39,43)(30,48,49,34)(31,44,41,38);(5,11)(7,8)(10,18)(12,13)(14,19)(15,16)(23,36)(27,46)(30,49)(31,41)(34,48)(35,51)(38,44)(45,50)

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, 5388);