Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C3×#7338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1316 |
| commutator | C24 | C2×C3×C8 |
| Frattini, cosocle | #379 | C22×C3 |
| prime radical | C23×C3×D4 | C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×C3 ◁ C24×C3 ◁ C3×#379 ◁ G | |
490 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 46 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)(4,5,7,11)(6,12,13,19,14,9,15,17)(8,10,16,18);(4,6)(5,9)(7,15)(8,14)(10,12)(11,17)(13,16)(18,19)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 326);