C₃×#10255 #10628862

order	768
nilpotency class	3
isoclinism	isoclinic class of #10254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#10255

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3245	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 582 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) = 384

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

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

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, 3243);