C₃×#13341 #10631948

order	768
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13341

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#97 ◁ G

594 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 23 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6329);