Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C27 |
| direct product | C3×#10712 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C22 | C2×C3×C4×C8 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
2038 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)(8,9)(10,11)(16,17,18,20,19,21,22,23);(5,7)(12,13,14,15);(9,11)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3700);