Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C24×C3 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,13,17)(10,18,19,36)(11,22,23,39)(12,27,25,16)(14,29,30,20)(15,32,33,26)(21,34,28,37)(24,31,38,35);(4,5)(6,7)(11,23)(15,33)(18,36)(20,29)(22,39)(24,38)(26,32)(34,37);(5,7)(8,10,12,28,13,19,25,21)(9,14,16,35,17,30,27,31)(11,24,26,29,23,38,32,20)(15,34,22,36,33,37,39,18)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6387);