C₃×#10225 #10628832

order	768
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#10225

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 574 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) = 43
transitive degree	μ_t(G) = 192

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

(1,2,3)(4,5)(6,9)(7,10)(8,11)(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);(5,9)(6,7)(10,11)(12,14)(13,19)(15,29)(16,28)(17,26)(18,27)(20,36)(21,35)(22,24)(23,34)(25,30)(31,42)(32,41)(33,37)(38,40)(39,43);(5,10)(6,7)(9,11)(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, 3213);