C₃×#10226 #10628833

order	768
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10226

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

488 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 192

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

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

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, 3214);