Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#22113 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(4,5,7,6)(9,12)(10,14)(17,20)(18,21)(22,23);(4,6,7,5)(8,9)(10,13)(11,12)(14,15)(16,17,19,20)(18,22,23,21);(9,12)(13,15)(16,18,19,23)(17,21,20,22);(8,10,11,14)(9,13,12,15)(16,17,19,20)(18,21,23,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15101);