Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#7106 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #331 |
| commutator | C42 | C2×C3×C8 |
| Frattini, cosocle | C43 | C22×C3 |
| socle | C2×C3 | #1270 |
| prime radical | C3×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
268 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 19 points:
(1,2,3)(4,5)(6,13,14,19,15,18,16,9)(7,10)(8,12)(11,17);(4,6)(5,9)(7,16)(8,15)(10,18)(11,13)(12,19)(14,17)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 94);