Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#37140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(4,5)(6,8)(7,10)(9,11)(12,13)(14,18)(15,17)(16,19)(23,30)(27,32)(29,34)(33,35);(5,6)(9,10)(12,13,15,17)(14,16,19,18)(21,25)(22,28)(27,32)(29,34);(4,6,8,5)(7,10,11,9)(12,13)(14,16)(15,17)(18,19)(20,21)(22,26)(23,27)(24,25)(28,31)(29,33)(30,32)(34,35);(4,7)(5,9)(6,10)(8,11)(12,14)(13,16)(15,19)(17,18)(20,22,24,28)(21,26,25,31)(23,29,30,34)(27,33,32,35);(20,23)(21,27)(22,29)(24,30)(25,32)(26,33)(28,34)(31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30128);