Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C25 |
| direct product | C3×#17789 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C23×C3×C8 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C2×C3×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×#41 ◁ G | |
734 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)(4,5,6,8,7,9,10,11)(13,17)(15,18)(19,23)(20,24)(22,25)(26,27);(12,13,18,24,16,20,15,17)(14,19,25,27,21,26,22,23);(12,14)(13,19)(15,22)(16,21)(17,23)(18,25)(20,26)(24,27);(14,21)(19,26)(22,25)(23,27)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10777);