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) = C2×C43 |
| direct product | C3×#14589 |
| 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 | |
1650 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)(8,9,11,10)(12,13,16,14)(15,18,19,17)(21,24)(25,26);(5,7)(13,14)(15,19)(20,21,23,24)(22,26,27,25);(4,5,6,7)(8,10,11,9)(12,14,16,13)(15,18,19,17)(20,21)(22,25)(23,24)(26,27);(12,15,16,19)(13,17,14,18)(20,22)(21,25)(23,27)(24,26)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 7577);