C₃×#9940 #10628547

order	768
nilpotency class	3
isoclinism	isoclinic class of #9938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#9940

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₃×#2539	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

768 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 27
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(5,8)(6,10)(9,11)(12,13)(14,20)(15,23)(16,19)(17,21)(18,25)(22,27)(24,26);(4,5,7,9)(6,11,10,8)(12,14)(13,17)(15,24)(16,22)(18,27)(19,26)(20,23)(21,25);(6,10)(8,11)(17,26)(20,27)

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, 2928);