Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C23 |
| direct product | C3×#7102 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C16 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| socle | C23×C3 | C22⋊C8 |
| prime radical | C22×C3×D4 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
288 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,16)(15,18)(17,19)(20,21,22,25,23,26,28,31,24,27,29,32,30,33,34,35);(4,6)(5,8)(7,11)(9,10)(13,16)(14,19)(17,18)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 90);