Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10922 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5)(6,7)(9,11)(16,17,20,22,19,18,21,23);(5,7)(12,13,14,15)(17,18)(22,23);(8,9,10,11)(16,18,20,23,19,17,21,22)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3910);