Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C24×C3×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25×C3 | D4 |
| commutator, Frattini, cosocle | C4 | C26×C3 |
| prime radical | C24×C3×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ C24×C3×C4 ◁ G | |
13310 conjugacy classes of subgroups; only the 6878 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 6878 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);(10,12)(11,15)(14,16)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56066);