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×#18246 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,9)(7,14)(10,16)(11,15)(13,18)(17,19)(25,31)(32,35);(4,5,8,11)(6,10,12,17)(7,15,14,9)(13,19,18,16)(20,21)(22,25)(23,26)(24,27)(28,32)(29,31)(30,33)(34,35);(4,6,8,12)(5,10,11,17)(7,13,14,18)(9,16,15,19)(20,22)(21,25)(23,28)(24,29)(26,32)(27,31)(30,34)(33,35);(6,12)(10,17)(13,18)(16,19)(20,23,24,30)(21,26,27,33)(22,28,29,34)(25,32,31,35)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11234);