C₃×#12518 #10631125

order	768
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₃×#12518

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₃×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

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

normal subgroups only

This group has 330 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) = 384

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

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

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, 5506);