Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×C8×#126 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C23 |
| commutator | C22 | C23×C3×C8 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,5,6,7,8,9,10,11);(13,17)(14,21)(19,26)(20,24)(22,25)(23,27);(12,13,18,24)(14,19,25,26)(15,17,16,20)(21,27,22,23);(12,14,15,22)(13,19,17,23)(16,21,18,25)(20,27,24,26)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14652);