C₃×#10216 #10628823

order	768
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10216

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

468 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 98 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)(4,5)(6,8)(7,10)(9,11)(12,13,18,23)(14,24,27,38)(15,28,31,36)(16,22,17,21)(19,32,34,26)(20,29,37,30)(25,33,41,35)(39,42,43,40);(5,8)(6,11)(9,10)(12,14)(13,19)(15,25)(16,32)(17,26)(18,27)(20,35)(21,38)(22,24)(23,34)(28,39)(29,42)(30,40)(31,41)(33,37)(36,43);(4,5,7,10)(6,9,11,8)(12,15,18,31)(13,20,23,37)(14,25,27,41)(16,30,17,29)(19,33,34,35)(21,28,22,36)(24,39,38,43)(26,42,32,40)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3204);