Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#37663 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2336 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)(8,9)(10,12)(11,14)(13,15)(16,17,21,25)(18,23,19,22)(20,24,28,31)(26,30,27,29);(5,7)(8,10,11,15)(9,12,14,13)(16,18,21,19)(17,22,25,23)(20,27,28,26)(24,30,31,29);(4,5)(6,7)(8,10)(9,13)(11,15)(12,14)(16,17)(18,22)(19,23)(20,24)(21,25)(26,29)(27,30)(28,31);(16,19,21,18)(17,23,25,22)(20,27,28,26)(24,30,31,29);(4,5)(6,7)(8,9)(10,13)(11,14)(12,15)(16,20)(17,24)(18,26)(19,27)(21,28)(22,29)(23,30)(25,31)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30651);