Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#7213 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #342 |
| commutator | C22×C4 | C3×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C22×C3 |
| prime radical | C25×C3 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ C22×C3×C22⋊C4 ◁ G | |
560 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5)(6,14,15,9)(7,11)(8,13)(10,16,18,19)(12,17);(5,9,11,10)(12,18,13,14)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 201);