Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C42 ◁ G | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,12)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19)(20,21)(22,30)(23,29)(24,26)(25,28)(27,32)(31,35)(33,34);(4,6,8,14)(5,9,11,12)(7,15,16,13)(10,18,19,17)(21,26)(22,31)(28,29)(32,33);(4,7,8,16)(5,10,11,19)(6,13,14,15)(9,17,12,18)(20,22,24,31)(21,27,26,34)(23,32,25,33)(28,30,29,35);(5,11)(9,12)(10,19)(17,18)(21,26)(27,34)(28,29)(30,35);(20,23,24,25)(21,28,26,29)(22,32,31,33)(27,30,34,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53854);