C₃×#10036 #10628643

order	768
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10036

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#392
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 528 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) = 27
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3024);