C₃×#13330 #10631937

order	768
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13330

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

558 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 43
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3)(4,5)(6,7)(12,13,18,23)(14,24,15,27)(16,30,35,43)(17,36,33,22)(19,28,20,25)(21,39,42,34)(26,41,29,40)(31,38,32,37);(5,7)(8,9)(10,11)(12,14,18,15)(13,19,23,20)(16,31,35,32)(17,29,33,26)(21,40,42,41)(22,38,36,37)(24,43,27,30)(25,39,28,34);(9,11)(12,15,18,14)(13,20,23,19)(16,32,35,31)(17,26,33,29)(21,41,42,40)(22,37,36,38)(24,30,27,43)(25,34,28,39)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6318);