Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
918 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)(5,6)(8,9)(10,13)(11,12)(14,15)(16,17,20,24)(18,21,25,28)(19,22,27,23)(26,29,31,30);(4,5,7,6)(9,12)(13,15)(16,18)(17,21)(19,26)(20,25)(22,29)(23,30)(24,28)(27,31);(4,6,7,5)(8,10,11,14)(9,13,12,15)(18,25)(21,28)(26,31)(29,30);(17,22)(19,27)(21,29)(23,24)(26,31)(28,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26024);