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×#37695 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(10,14)(12,15)(16,17,20,23)(18,24,26,21)(19,22,28,27)(25,30,31,29);(5,7)(8,9,11,13)(10,15,14,12)(16,17)(18,21)(19,27)(20,23)(22,28)(24,26)(25,30)(29,31);(4,5)(6,7)(8,10,11,14)(9,12,13,15);(16,18,20,26)(17,21,23,24)(19,25,28,31)(22,29,27,30);(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,19)(17,22)(18,25)(20,28)(21,29)(23,27)(24,30)(26,31)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30683);