Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C42 |
| direct product | C3×#14932 |
| 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 | |
1938 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)(9,11)(12,13)(14,18)(15,19)(16,17)(22,26)(24,27);(5,7)(9,11)(13,17)(14,19)(20,21,23,25)(22,24,26,27);(4,5)(6,7)(8,9,10,11)(12,14,16,19)(13,18,17,15)(20,21)(22,24)(23,25)(26,27);(12,15,16,18)(13,19,17,14)(20,22)(21,24)(23,26)(25,27)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7920);