Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(14,22)(15,25)(16,27)(18,30)(20,35)(21,34)(23,40)(24,39)(26,29)(28,38)(31,42)(32,43)(33,41)(36,37);(4,6)(5,8)(7,11)(9,10)(12,13)(14,18)(15,20)(16,21)(17,19)(22,30)(23,32)(24,31)(25,35)(26,36)(27,34)(28,33)(29,37)(38,41)(39,42)(40,43);(6,11)(8,10)(12,14,17,24)(13,18,19,31)(15,26,28,40)(16,29,27,23)(20,36,33,43)(21,37,34,32)(22,38,39,25)(30,41,42,35);(5,9)(8,10)(13,19)(18,31)(20,33)(21,34)(30,42)(32,37)(35,41)(36,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26755);