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×#37159 |
| 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 | |
1500 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)(5,9)(7,10)(13,16)(17,19)(21,22)(23,30)(24,32)(33,34);(5,9)(6,11)(20,21,25,22)(23,28,30,26)(24,27,32,29)(31,34,35,33);(4,5)(6,10)(7,11)(8,9)(12,13)(14,17)(15,16)(18,19)(20,22,25,21)(23,28,30,26)(24,29,32,27)(31,34,35,33);(4,6,8,11)(5,10,9,7)(20,23,25,30)(21,26,22,28)(24,31,32,35)(27,33,29,34);(4,7,8,10)(5,11,9,6)(12,14,15,18)(13,17,16,19)(20,24)(21,27)(22,29)(23,31)(25,32)(26,33)(28,34)(30,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 30147);