Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#20655 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C22⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#117 ◁ G | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,7)(8,9,12,16,13,11,20,15)(10,14,19,22,18,17,23,21);(4,5,6,7)(8,10)(9,14)(11,17)(12,19)(13,18)(15,21)(16,22)(20,23);(8,11)(9,13)(10,17)(12,15)(14,18)(16,20)(19,21)(22,23);(10,18)(14,17)(19,23)(21,22)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13643);