Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C32 · Schur multiplier (H2) = C22 |
| direct product | C3×#7362 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22⋊C4 |
| commutator | C22 | C2×C3×C32 |
| Frattini, cosocle | C22×C16 | C22×C3 |
| socle | C22×C3 | C22⋊C16 |
| prime radical | C23×C3 | C2×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C16 ◁ G | |
138 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13,14,18,15,19,22,28,16,20,23,29,25,31,34,38,17,21,24,30,26,32,35,39,27,33,36,40,37,41,42,43);(4,6,7,11)(5,8,9,10)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 350);