C₃×#10689 #10629296

order	768
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#10689

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#124
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#400
prime radical	C₂³×C₃	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

248 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,5,7,8)(6,9,11,10)(12,13,16,15,17,19,21,25)(14,20,22,27,23,24,26,18);(4,6,7,11)(5,9,8,10)(12,14,21,22,17,23,16,26)(13,18,25,20,19,27,15,24);(5,8)(6,10)(9,11)(12,15,16,19,17,25,21,13)(14,20,22,27,23,24,26,18)

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, 3677);