Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×#1355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #329 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | #97 | C23×C3 |
| prime radical | C2×C3×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#97 ◁ G | |
366 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 110 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);(3,4,5,10,6,8,14,16,7,11,13,18,15,9,12,17);(4,8)(5,12)(6,15)(9,11)(10,18)(13,14)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 4897);