Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C3×#2501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
4222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(19,20,21);(4,5)(8,9)(11,12,15,17)(13,14,16,18);(3,4,6,5)(7,8,10,9)(11,13)(12,14)(15,16)(17,18);(7,8,10,9)(11,14)(12,16)(13,17)(15,18);(3,5,6,4)(7,9,10,8)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27744);