Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#41415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(8,9,12,11)(10,14,15,13)(16,17)(18,24)(19,23)(20,21)(22,26)(25,30)(27,28)(29,31);(9,11)(13,14)(17,21)(22,24)(23,28)(29,30);(4,5,6,7)(10,15)(13,14)(16,17)(18,22)(19,23)(20,21)(24,26)(25,29)(27,28)(30,31);(8,10)(9,13)(11,14)(12,15)(16,18)(17,22)(19,25)(20,26)(21,24)(23,29)(27,31)(28,30);(8,11,12,9)(10,14,15,13)(16,19,20,27)(17,23,21,28)(18,25,26,31)(22,29,24,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34403);