Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×C22⋊#401 |
| 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 | |
1650 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)(8,9,11,13)(10,12,14,15)(16,17,20,21)(18,23,22,19)(24,25)(26,29)(27,31)(28,30);(10,14)(12,15)(17,21)(18,22)(24,26,28,31)(25,29,30,27);(8,10)(9,12)(11,14)(13,15)(16,18,20,22)(17,19,21,23)(24,26)(25,27)(28,31)(29,30);(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,19)(17,22)(18,21)(20,23)(24,27)(25,26)(28,29)(30,31)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7815);