Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#15705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1538 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)(9,12)(10,14)(16,17,20,18)(19,22,23,21)(25,26)(27,31);(17,18)(19,23)(24,25,28,26)(27,30,31,29);(8,9)(10,13)(11,12)(14,15)(16,18,20,17)(19,22,23,21)(24,26,28,25)(27,30,31,29);(4,5)(6,7)(8,10,11,14)(9,13,12,15)(16,19)(17,21)(18,22)(20,23)(24,27,28,31)(25,29,26,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8693);