Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#37675 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2922 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)(9,11)(12,13,16,19)(14,20,22,17)(15,18,24,23)(21,26,27,25);(5,7)(12,13)(14,17)(15,23)(16,19)(18,24)(20,22)(21,26)(25,27);(4,5)(6,7)(8,9,10,11);(12,14,16,22)(13,17,19,20)(15,21,24,27)(18,25,23,26);(4,5)(6,7)(12,15)(13,18)(14,21)(16,24)(17,25)(19,23)(20,26)(22,27)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30663);