Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#29695 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1024 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)(9,10)(13,14)(16,17,20,24)(18,25,21,22)(19,27,28,31)(23,29,30,26);(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,18,20,21)(17,22,24,25)(19,26,28,29)(23,27,30,31);(5,7)(8,10,12,9)(11,13,15,14)(16,19,20,28)(17,23,24,30)(18,26,21,29)(22,27,25,31);(8,11,12,15)(9,13,10,14)(19,28)(23,30)(26,29)(27,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22683);