C₃×#7825 #10626432

order	768
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#7825

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#2682	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 516 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) = 384

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

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

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 813);