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×#33770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
576 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)(12,13,16,22)(14,25,27,19)(15,24,18,21)(17,32,31,35)(20,38,39,28)(23,30,41,34)(26,36,29,40)(33,42,43,37);(4,6)(5,8)(7,11)(9,10)(12,14,16,27)(13,19,22,25)(15,29,18,26)(17,28,31,38)(20,32,39,35)(21,40,24,36)(23,37,41,42)(30,43,34,33);(6,11)(8,10)(12,14)(13,20)(15,26)(16,27)(17,33)(18,29)(19,35)(21,37)(22,39)(23,40)(24,42)(25,32)(28,34)(30,38)(31,43)(36,41);(5,9)(8,10)(12,15,16,18)(13,21,22,24)(14,26,27,29)(17,30,31,34)(19,36,25,40)(20,37,39,42)(23,35,41,32)(28,33,38,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26758);