Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×#3054 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C23×C3 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G | |
2918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 25 points:
(1,2)(23,24,25);(3,4)(5,6)(7,8)(9,16)(10,14)(11,20)(12,15)(13,19)(17,22)(18,21);(7,9,11,19,12,17,10,18)(8,13,14,16,15,21,20,22);(4,6)(9,17)(13,21)(16,22)(18,19);(7,10,12,11)(8,14,15,20)(9,18,17,19)(13,16,21,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53429);