Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C3×#60144 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
2778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5)(6,7)(8,9,13,16,12,15,21,23)(10,17,19,20,18,14,22,11);(5,7)(8,10)(9,14)(11,16)(12,18)(13,19)(15,17)(20,23)(21,22);(8,11,13,17,12,20,21,14)(9,10,16,19,15,18,23,22);(9,15)(11,20)(14,17)(16,23);(9,15)(10,18)(16,23)(19,22)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53132);