Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C3×#1412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #354 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | #127 | C23×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#127 ◁ G | |
390 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 390 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)(5,12,14,18)(6,10,16,9)(7,11)(8,13,17,15);(3,5,7,13)(4,8)(6,15,16,14)(9,12)(10,17)(11,18)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6581);