C₃×#12504 #10631111

order	768
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#12504

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 43
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 3 generators on 43 points:

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

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5492);