Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C3×C4 |
| Frattini, cosocle | C4⋊Q8 | C23×C3 |
| prime radical | C2×C3×2−1+4 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×C4⋊Q8 ◁ G | |
484 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(19,20,21);(3,4,6,10,8,5,12,14)(7,11,17,9,18,15,16,13);(3,5)(4,8)(6,9)(7,11)(10,17)(12,13)(14,16)(15,18)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6559);