Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#29690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,10)(13,14)(16,17)(18,23)(19,21)(20,22);(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(17,21)(22,23);(5,7)(8,10,12,9)(11,13,15,14)(17,22)(18,20)(21,23);(8,11,12,15)(9,13,10,14)(16,18,19,20)(17,23,21,22)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22678);