Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C42 |
| direct product | C3×#14503 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1558 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)(4,5,6,7)(9,12)(10,14)(16,17,20,18)(19,22,23,21)(25,28)(29,30);(9,12)(13,15)(17,18)(19,23)(24,25,27,28)(26,30,31,29);(8,9,11,12)(10,13,14,15)(16,18,20,17)(19,22,23,21)(24,25)(26,29)(27,28)(30,31);(8,10)(9,13)(11,14)(12,15)(16,19,20,23)(17,21,18,22)(24,26)(25,29)(27,31)(28,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7491);