C₃×#12158 #10630765

order	768
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#12158

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₈ ◁ G

434 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,11)(12,13)(14,17)(15,24)(16,20)(18,21)(19,22)(23,26)(25,27);(4,5)(6,8)(7,9)(10,11)(13,17)(14,21)(15,22)(18,19)(20,26)(23,27)(24,25);(4,6,7,10)(5,8,9,11)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5146);