Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#35316 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(5,7)(9,11)(12,13,16,19)(14,17,22,20)(15,23,24,18)(21,26,27,25);(8,9)(10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,24)(25,27);(4,5,6,7)(12,14)(13,17)(15,21)(16,22)(18,25)(19,20)(23,26)(24,27);(12,14)(13,17)(15,21)(16,22)(18,25)(19,20)(23,26)(24,27);(12,15,16,24)(13,18,19,23)(14,21,22,27)(17,25,20,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28304);