Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#7049 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #95 |
| commutator | C22×C4 | C3×C42 |
| Frattini, cosocle | C43 | C22×C3 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C43 ◁ G | |
304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5,7,10,8,11,6,9)(12,13)(14,16)(15,18)(17,19)(21,25,26,27);(5,9,11,10)(13,16)(14,19)(17,18)(20,21,23,25,24,26,22,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 37);