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×#37127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C3×#2688 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1566 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)(6,7)(14,18)(16,19)(23,30)(27,32)(29,34)(33,35);(4,5,8,9)(6,10,7,11)(12,13,15,17)(14,16,18,19)(21,25)(22,28)(27,32)(29,34);(4,6,8,7)(5,10,9,11)(12,14)(13,16)(15,18)(17,19)(20,21)(22,26)(23,27)(24,25)(28,31)(29,33)(30,32)(34,35);(4,5)(6,10)(7,11)(8,9)(20,22,24,28)(21,26,25,31)(23,29,30,34)(27,33,32,35);(4,7,8,6)(5,11,9,10)(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, 30115);