Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#36728 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,8,9)(6,11,10,7)(14,18)(17,19)(22,26)(23,27);(5,9)(6,10)(13,16)(17,19)(20,21,24,25)(22,27,26,23);(12,13)(14,17)(15,16)(18,19)(20,22)(21,23)(24,26)(25,27);(4,6,8,10)(5,7,9,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,23)(24,25)(26,27);(4,7)(5,10)(6,9)(8,11)(12,14)(13,17)(15,18)(16,19)(20,23)(21,22)(24,27)(25,26)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29716);