Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#60049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1634 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)(14,18)(15,19)(20,21,25,23)(22,24,27,26);(6,10)(9,11)(12,13,17,16)(14,19,18,15)(20,22,25,27)(21,26,23,24);(4,5,8,7)(6,9,10,11)(12,14)(13,15)(16,19)(17,18)(20,23,25,21)(22,24,27,26);(4,6,8,10)(5,9,7,11)(12,15,17,19)(13,18,16,14);(4,7,8,5)(6,11,10,9)(12,16,17,13)(14,19,18,15)(20,24)(21,22)(23,27)(25,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53037);