Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25772 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#3496 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,6)(8,9,12,15)(10,13,17,21)(11,14,19,22)(16,23,18,20);(4,5,7,6)(8,10)(9,13)(11,18)(12,17)(14,20)(15,21)(16,19)(22,23);(8,11)(9,14)(10,16,17,18)(12,19)(13,20,21,23)(15,22);(4,6,7,5)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18760);