Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C3×#34289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#547 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
3032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2,3)(4,5,7,6)(9,11)(12,13)(14,20)(15,23)(16,17)(18,22)(19,24)(21,25)(26,27);(13,17)(18,20)(19,23)(25,26);(8,9)(10,11)(12,14,16,22)(13,18,17,20)(15,21,24,27)(19,25,23,26);(12,13)(14,18)(15,19)(16,17)(20,22)(21,25)(23,24)(26,27);(4,6,7,5)(12,15)(13,19)(14,21)(16,24)(17,23)(18,25)(20,26)(22,27)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27277);