Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3)(9,15)(16,19)(20,21,25,29)(22,30,33,43)(23,27,34,41)(24,36,38,28)(26,37,40,32)(31,44,45,51)(35,48,47,42)(39,49,50,46);(4,5)(6,9)(7,10)(8,11)(12,16)(13,15)(14,17)(18,19)(20,22,25,33)(21,26,29,40)(23,31,34,45)(24,37,38,32)(27,39,41,50)(28,43,36,30)(35,49,47,46)(42,51,48,44);(4,6)(5,9)(7,12)(8,13)(10,16)(11,15)(14,18)(17,19)(20,23,25,34)(21,27,29,41)(22,31,33,45)(24,35,38,47)(26,39,40,50)(28,42,36,48)(30,44,43,51)(32,46,37,49);(4,7,8,14)(5,10,11,17)(6,12,13,18)(9,16,15,19)(23,34)(27,41)(31,45)(35,47)(39,50)(42,48)(44,51)(46,49)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11235);