Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#21676 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C23 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,11)(8,9)(12,13)(14,16)(15,17)(18,19)(20,21,26,29,25,28,23,24)(22,30,31,32,33,27,34,35);(5,9)(10,11)(12,14,15,18)(13,16,17,19)(20,22)(21,27)(23,34)(24,32)(25,33)(26,31)(28,30)(29,35);(4,6,8,7)(5,10,9,11)(13,17)(16,19)(20,23,25,26)(21,24,28,29)(22,31,33,34)(27,35,30,32);(4,7,8,6)(5,11,9,10)(20,24,23,28,25,29,26,21)(22,32,34,30,33,35,31,27)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14664);