C₃×#10005 #10628612

order	768
nilpotency class	3
isoclinism	isoclinic class of #10004
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10005

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3109	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 566 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) = 192

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

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

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, 2993);