Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#21985 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
806 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)(18,25)(19,27)(20,24)(21,22)(23,28)(26,31)(29,30);(8,9)(10,13)(11,12)(14,15)(16,18,20,21)(17,22,24,25)(19,28,29,26)(23,27,31,30);(9,12)(13,15)(16,19,21,28,20,29,18,26)(17,23,25,27,24,31,22,30);(8,10,11,14)(9,13,12,15)(17,24)(22,25)(23,31)(27,30)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14973);