Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C3×#20358 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(8,9,11,15,12,13,18,20)(10,14,17,21,16,19,22,23);(4,5,6,7)(9,13)(11,18)(14,19)(17,22);(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23);(10,16)(14,19)(17,22)(21,23)246 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13346);