Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#16601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(5,7)(9,11)(12,13,15,17)(14,18,19,16)(20,21,25,23)(22,27,26,24);(12,13)(14,16)(15,17)(18,19)(20,22,25,26)(21,24,23,27);(8,9)(10,11)(12,13)(14,18)(15,17)(16,19)(20,23,25,21)(22,24,26,27);(4,5)(6,7)(8,9)(10,11)(12,14)(13,16)(15,19)(17,18)(20,24)(21,26)(22,23)(25,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9589);