C₃×#10388 #10628995

order	768
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10388

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

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

normal subgroups only

This group has 594 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) = 27
transitive degree	μ_t(G) = 192

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

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

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, 3376);