Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×C4×#125 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| socle, prime radical | C24×C3 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
2622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(3,4,5,6);(9,13)(11,14)(15,16,18,20)(17,19,21,22);(7,8,10,12)(9,11,13,14)(15,16)(17,19)(18,20)(21,22);(7,9)(8,11)(10,13)(12,14)(15,17,18,21)(16,19,20,22)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27022);