Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#21248 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×#121 |
| prime radical | C22×C3×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9,13,19,14,18,12,17)(10,15,21,29,22,31,27,30)(11,23,25,33,26,32,24,16)(20,34,35,28,36,38,37,39);(4,5,6,7)(8,10,13,27,14,22,12,21)(9,15,17,29,18,31,19,30)(11,20,25,37,26,36,24,35)(16,28,32,38,33,39,23,34);(8,11,12,24,14,26,13,25)(9,16,19,23,18,33,17,32)(10,20,21,35,22,36,27,37)(15,28,30,34,31,39,29,38);(8,12,14,13)(9,17,18,19)(10,21,22,27)(11,24,26,25)(15,29,31,30)(16,32,33,23)(20,35,36,37)(28,38,39,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14236);