Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#23581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1122 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)(8,9)(10,13)(11,12)(14,15)(16,17)(18,27)(19,23)(20,25)(21,26)(22,30)(24,29)(28,31);(4,5)(6,7)(9,12)(13,15)(16,18)(17,23)(19,21)(20,29)(22,28)(24,30)(25,31)(26,27);(5,7)(16,19,21,18)(17,24,26,31)(20,29,22,28)(23,25,27,30);(8,10,11,14)(9,13,12,15)(16,20,21,22)(17,25,26,30)(18,28,19,29)(23,24,27,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16569);