Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C8 · Schur multiplier (H2) = C27 |
| direct product | C22×C3×#324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C22⋊C4 |
| commutator | C22 | C23×C3×C8 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C3×C4 ◁ G | |
1806 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)(21,22,23);(3,4);(5,6)(7,11,12,9)(8,10)(13,14,15,17,16,18,19,20);(5,7,8,12)(6,9,10,11)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10332);