Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C3×C4×#124 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| socle | C24×C3 | C24 |
| prime radical | C23×C3×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
3790 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);(3,4,5,6);(8,9)(13,17)(15,18);(7,8,10,9)(11,12,14,16)(13,15,17,18);(7,9,10,8)(11,13)(12,15)(14,17)(16,18)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27020);