Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#35313 |
| 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 | |
1658 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)(6,10)(8,11)(13,16)(14,18)(20,21,24,27)(22,25,30,28)(23,31,32,26)(29,34,35,33);(4,5,7,9)(6,8,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,24,27)(22,28,30,25)(23,26,32,31)(29,34,35,33);(4,6)(5,8)(7,10)(9,11)(12,14,15,18)(13,17,16,19)(20,21)(22,25)(23,26)(24,27)(28,30)(29,33)(31,32)(34,35);(20,22)(21,25)(23,29)(24,30)(26,33)(27,28)(31,34)(32,35);(20,23,24,32)(21,26,27,31)(22,29,30,35)(25,33,28,34)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 28301);