Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#37133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2168 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)(9,13)(11,15)(16,17)(18,24)(19,26)(20,28)(21,22)(23,29)(25,27)(30,31);(9,13)(10,14)(17,22)(19,24)(23,28)(27,30);(4,5,6,7)(8,9,12,13)(10,11,14,15)(16,18)(17,19)(20,25)(21,26)(22,24)(23,27)(28,30)(29,31);(8,10,12,14)(9,11,13,15)(16,19)(17,18)(20,27)(21,24)(22,26)(23,25)(28,31)(29,30);(8,11)(9,14)(10,13)(12,15)(16,20,21,29)(17,23,22,28)(18,25,26,31)(19,27,24,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30121);