Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C3×D4×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C3×D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5,6,7);(5,7);(8,9,11,14,12,15,18,21)(10,13,17,20,16,19,22,23);(8,10)(9,13)(11,17)(12,16)(14,20)(15,19)(18,22)(21,23);(10,16)(13,19)(17,22)(20,23)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53133);