C₃×#29231 #10647838

order	768
nilpotency class	3
isoclinism	isoclinic class of #24986
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#29231

Subgroups

	subgroup	G/H
center	C₃×C₄	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂×C₃	#2571
prime radical	C₂³×C₃	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ G

404 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 96

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

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

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, 22219);