C₃×#11679 #10630286

order	768
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#11679

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#102
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₂³×C₃	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

318 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 122 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)(4,5,7,10,8,6,11,9)(12,13)(14,23,26,40)(15,19)(16,30)(17,22)(18,25,34,41)(20,28)(21,29)(24,39,27,36)(31,38,42,33)(32,35)(37,43);(4,6)(5,8)(7,9)(10,11)(12,14,17,26)(13,18,22,34)(15,24,29,27)(16,31,32,42)(19,33,21,38)(20,36,37,39)(23,30,40,35)(25,28,41,43);(5,6)(9,10)(12,15,17,29)(13,19,22,21)(14,24,26,27)(16,28,32,43)(18,33,34,38)(20,35,37,30)(23,39,40,36)(25,31,41,42)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4667);