Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#18002 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C3×#449 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×M16 ◁ G | |
610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,13)(7,12)(8,16)(9,11)(10,14)(15,19)(17,18)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(4,6)(5,10)(7,14)(8,17)(9,15)(11,18)(12,19)(13,16);(5,11)(10,18)(12,16)(13,19)(20,22,23,29,24,28,30,34)(21,25,26,32,27,31,33,35);(4,7,9,8)(5,12,11,16)(6,14,15,17)(10,19,18,13)(22,28)(25,31)(29,34)(32,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 10990);