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×#41431 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1518 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)(5,8)(6,10)(14,18)(16,19)(20,21,25,26)(22,24,28,34)(23,30,33,27)(29,35,31,32);(4,5,7,8)(6,9,10,11)(12,13,15,17)(14,16,18,19)(21,26)(24,34)(27,30)(32,35);(4,5)(6,9)(7,8)(10,11)(12,14,15,18)(13,16,17,19)(20,22,25,28)(21,24,26,34)(23,31,33,29)(27,35,30,32);(4,6,7,10)(5,9,8,11)(20,23,25,33)(21,27,26,30)(22,29,28,31)(24,32,34,35);(20,24)(21,28)(22,26)(23,32)(25,34)(27,31)(29,30)(33,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 34419);