Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#2932 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
2806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(3,4)(5,6)(7,8,11,15,12,14,20,21)(9,13,17,10,16,18,22,19);(4,6)(7,9)(8,13)(10,15)(11,17)(12,16)(14,18)(19,21)(20,22);(7,10,11,18,12,19,20,13)(8,9,15,17,14,16,21,22);(8,14)(9,16)(15,21)(17,22)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53124);