C₃×#29295 #10647902

order	768
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#29295

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376 ◁ G

630 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 35
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3)(4,5,8,11)(6,10,7,9)(20,21)(22,29)(23,27)(24,26)(25,30)(28,32)(31,34)(33,35);(4,6,8,7)(5,9,11,10)(12,13,15,17)(14,18,19,16)(20,22)(21,25)(23,30)(24,31)(26,33)(27,29)(28,34)(32,35);(12,14,15,19)(13,16,17,18)(21,26)(24,32)(25,33)(27,28)(29,34)(31,35);(4,7,8,6)(5,10,11,9)(21,27)(22,30)(26,28)(31,35)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 22283);