C₃×#32901 #10651508

order	768
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₃×#32901

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃
prime radical	C₂×C₃×#585	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

3706 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9,13,12)(10,14,18,17)(11,15,20,19)(16,22,23,21);(8,10)(9,14)(11,16)(12,17)(13,18)(15,21)(19,22)(20,23);(8,11)(9,15)(10,16)(12,19)(13,20)(14,21)(17,22)(18,23);(5,7)(8,12,13,9)(10,17,18,14)(11,19,20,15)(16,22,23,21)

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 25889);