Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#12364 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C3×#3291 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
358 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 78 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)(5,9,11,19)(6,13)(7,16)(10,17,12,14)(15,18);(6,13)(7,16)(9,17)(10,12)(14,19)(15,18);(4,5)(6,14)(7,10)(8,11)(9,18)(12,16)(13,19)(15,17)102 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10708944, C3×#489347, #72215228, … |
| maximal normal subgroup of | C3×#489347, #72215228, D3×#12364 |
| maximal quotient of | C3×#10708944, C3×#136361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5352);