C₃×#13356 #10631963

order	768
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13356

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#2021
commutator, Frattini, cosocle	C₂×#41	C₂³×C₃
prime radical	C₃×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×#41 ◁ G

676 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3)(4,5,10,15)(6,16,20,38)(7,21,24,40)(8,25,28,13)(9,29,31,14)(11,26,34,18)(12,30,37,23)(17,43,46,67)(19,49,51,42)(22,54,56,53)(27,39,61,58)(32,63,35,66)(33,48,62,59)(36,55,57,60)(41,50,64,45)(44,52,47,65);(4,6)(5,11)(7,17)(8,26)(9,19)(10,20)(12,35)(13,38)(14,33)(15,34)(16,25)(18,28)(21,52)(22,47)(23,45)(24,46)(27,59)(29,62)(30,50)(31,51)(32,37)(36,64)(39,49)(40,65)(41,57)(42,58)(43,54)(44,56)(48,61)(53,67)(55,66)(60,63);(4,7,10,24)(5,12,15,37)(6,17,20,46)(8,22,28,56)(9,30,31,23)(11,32,34,35)(13,36,25,57)(14,40,29,21)(16,41,38,64)(18,47,26,44)(19,50,51,45)(27,60,61,55)(33,52,62,65)(39,54,58,53)(42,67,49,43)(48,63,59,66)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6344);