C₃×#19096 #10637703

order	768
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#19096

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 906 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) = 192

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

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 12084);