Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#14832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(5,7)(8,9,12,10)(11,14,15,13)(16,17,19,21)(18,20,22,23)(24,25,29,28)(26,31,27,30);(9,10)(13,14)(18,22)(20,23)(24,26,29,27)(25,30,28,31);(8,10,12,9)(11,14,15,13)(16,18)(17,20)(19,22)(21,23)(24,27,29,26)(25,31,28,30);(4,5)(6,7)(8,11,12,15)(9,13,10,14)(16,17)(18,20)(19,21)(22,23)(24,28,29,25)(26,31,27,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7820);