Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C4 |
| direct product | C3×#7372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #353 |
| commutator | C42 | C2×C3×C8 |
| Frattini, cosocle | #382 | C22×C3 |
| prime radical | C23×C3 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#382 ◁ G | |
178 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3)(4,5,6,8,7,9,10,11)(12,13,15,20,17,22,14,23)(16,19,24,18,25,21,26,27);(12,14,16,24,17,15,25,26)(13,18,19,20,22,27,21,23)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 360);