C₃×#12088 #10630695

order	768
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#12088

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#442
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#322 ◁ G

198 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 90 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)(4,5,8,6,7,9,11,10)(12,13,16,20,18,22,17,33)(14,23,26,41,24,42,27,38)(15,28,31,40,30,39,32,21)(19,34,35,25,36,29,37,43);(5,9)(6,10)(12,14,18,24)(13,19,22,36)(15,29,30,34)(16,27,17,26)(20,37,33,35)(21,38,40,41)(23,28,42,39)(25,32,43,31);(4,6,8,9,7,10,11,5)(13,20,22,33)(14,24)(15,30)(19,35,36,37)(21,39,40,28)(23,38,42,41)(26,27)(31,32)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5076);