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×#36745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2184 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)(10,14)(13,15)(16,17)(18,19)(20,24)(21,22)(23,27)(25,26)(28,29)(30,31);(9,12)(13,15)(17,22)(18,23)(24,29)(25,30);(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,18,21,23)(17,19,22,27)(20,25,28,30)(24,26,29,31);(8,9)(10,13)(11,12)(14,15)(16,19)(17,23)(18,22)(20,26)(21,27)(24,30)(25,29)(28,31);(8,10)(9,13)(11,14)(12,15)(16,20,21,28)(17,24,22,29)(18,25,23,30)(19,26,27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29733);