C₃×#12530 #10631137

order	768
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₃×#12530

Subgroups

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

284 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 94 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)(6,10)(9,11)(12,13,15,18)(14,22,24,36)(16,29,27,42)(17,30,28,21)(19,33,31,25)(20,35,32,23)(26,41,40,38)(34,39,37,43);(4,5,6,8,7,9,10,11)(13,18)(19,31)(20,32)(21,30)(22,36)(29,42)(34,37)(38,41);(5,9)(8,11)(12,14,23,39,17,26,27,33,15,24,35,43,28,40,16,25)(13,19,29,38,21,34,32,36,18,31,42,41,30,37,20,22)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5518);