C₃×#8735 #10627342

order	768
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#8735

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#381
commutator	C₂³	C₂²×C₃×C₈
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

514 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 166 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)(4,5,6,8,7,9,10,11)(12,13)(14,20,22,17)(15,23)(16,19)(18,24)(21,25,27,26);(12,14,16,22)(13,17,19,20)(15,21,24,27)(18,25,23,26);(12,15)(13,18,19,23)(14,21)(16,24)(17,25,20,26)(22,27)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1723);