Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#14513 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
1918 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)(5,6)(8,9,10,11)(13,16)(14,18)(20,21)(22,24)(23,25)(26,27);(13,16)(17,19)(21,22)(23,27);(4,5,7,6)(12,13)(14,17)(15,16)(18,19)(20,22,24,21)(23,26,27,25);(4,6,7,5)(12,14)(13,17)(15,18)(16,19)(20,23,24,27)(21,25,22,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7501);