Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C4 · Schur multiplier (H2) = C214 |
| direct product | C24×C3×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C3×C4 | C22 |
| commutator | C2 | C25×C3×C4 |
| Frattini, cosocle | C4 | C26×C3 |
| socle | C25×C3 | C2×C4 |
| prime radical | C26×C3 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3×C4 ◁ G | |
13310 conjugacy classes of subgroups; only the 11166 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 11166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 19 points:
(1,2)(17,18,19);(3,4);(5,6);(7,8);(9,10,11,12,13,14,15,16);(10,14)(12,16)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56060);