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×#25773 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,6)(8,9)(10,16)(11,14)(12,15)(13,18)(17,21)(19,23)(20,22);(4,5,7,6)(8,10,12,18)(9,13,15,16)(11,19,20,17)(14,21,22,23);(8,11)(9,14)(10,17,18,19)(12,20)(13,21,16,23)(15,22);(4,6,7,5)(9,15)(13,16)(14,22)(21,23)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18761);