Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C3×#35305 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C23×C3×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
5230 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)(5,7)(9,11)(12,13,16,17)(14,18,20,24)(15,22,23,19)(21,27,26,25);(8,9)(10,11)(13,17)(14,20)(19,22)(21,26);(4,5)(6,7)(8,9)(10,11)(12,13)(14,18)(15,19)(16,17)(20,24)(21,25)(22,23)(26,27);(12,14)(13,18)(15,21)(16,20)(17,24)(19,25)(22,27)(23,26);(12,15,16,23)(13,19,17,22)(14,21,20,26)(18,25,24,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28293);