C₃×#10382 #10628989

order	768
nilpotency class	3
isoclinism	isoclinic class of #10382
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#10382

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂×C₃×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

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

normal subgroups only

This group has 446 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) = 96

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

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3370);