Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C42 |
| direct product | C3×#20760 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C22×D4 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G | |
950 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)(8,9)(10,13)(11,12)(14,15)(16,17)(18,24)(19,23)(20,25)(21,26)(22,27)(28,31)(29,30);(9,12)(13,15)(16,18,21,28,22,29,20,19)(17,23,25,30,27,31,26,24);(8,10,11,14)(9,13,12,15)(16,19,20,29,22,28,21,18)(17,24,26,31,27,30,25,23);(16,20,22,21)(17,25,27,26)(18,19,29,28)(23,30,31,24)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13748);