C₃×#12520 #10631127

order	768
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₃×#12520

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₃×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

316 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 384

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

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5508);