Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#40564 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1478 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)(12,13,16,14)(15,18,19,17)(20,21)(22,26)(23,30)(24,25)(27,32)(28,35)(29,33)(31,34);(4,5)(6,10)(7,8)(9,11)(12,14,16,13)(15,18,19,17)(21,25)(26,33)(27,30)(34,35);(5,8)(9,10)(12,15)(13,17)(14,18)(16,19)(20,22,24,29)(21,26,25,33)(23,31,32,28)(27,35,30,34);(4,5)(6,9)(7,8)(10,11)(20,23,24,32)(21,27,25,30)(22,28,29,31)(26,34,33,35);(4,6)(5,9)(7,11)(8,10)(20,21)(22,26)(23,27)(24,25)(28,34)(29,33)(30,32)(31,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 33552);