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) = C22×C4 |
| direct product | C3×#13568 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C3×C4 |
| Frattini, cosocle | #124 | C23×C3 |
| socle | C2×C3 | C2×AΣL(1,𝔽16) |
| prime radical | C3×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#532 ◁ G | |
498 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 19 points:
(1,2,3)(4,5,7,11)(6,9,14,16)(8,13,18,19)(10,15,12,17);(4,5)(6,9)(7,16)(8,13)(10,18)(11,17)(12,15)(14,19);(4,6,8,15)(5,9,13,12)(7,14,18,17)(10,11,16,19)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6556);