C₃×#8531 #10627138

order	768
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#8531

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

728 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 150 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)(4,5,7,9,8,6,10,11)(12,13,17,19,16,15,23,24)(14,18,22,26,21,20,25,27);(4,6)(5,8)(7,9)(10,11)(12,14,16,21)(13,18,15,20)(17,25,23,22)(19,27,24,26);(7,10)(9,11)(12,15)(13,16)(14,20)(17,19)(18,21)(22,26)(23,24)(25,27)

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, 1519);