Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#16599 |
| 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 | |
1550 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)(8,9,12,11)(10,14,15,13)(17,20)(18,22)(24,25,27,29)(26,28,30,31);(16,17)(18,21)(19,20)(22,23)(24,26,27,30)(25,28,29,31);(8,10,12,15)(9,13,11,14)(16,17)(18,23)(19,20)(21,22)(24,26)(25,28)(27,30)(29,31);(4,5)(6,7)(8,11,12,9)(10,14,15,13)(16,18,19,22)(17,21,20,23)(24,25)(26,28)(27,29)(30,31)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 9587);