C₃×#8774 #10627381

order	768
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#8774

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#381
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

384 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 114 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)(4,5,7,10,8,6,11,9)(12,13)(14,18,15,23)(16,21)(17,20)(19,24,27,22)(25,26);(12,14,16,15)(13,18,21,23)(17,22,25,24)(19,26,27,20);(4,6)(5,8)(7,9)(10,11)(12,15,16,14)(13,19)(17,24,25,22)(18,26)(20,23)(21,27)

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1762);