Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#25820 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(8,9,12,15)(10,14,11,13)(18,24)(19,26)(21,27)(23,29)(25,30)(28,31);(4,5)(6,7)(8,10,12,11)(9,13,15,14)(16,17)(18,21)(19,23)(20,22)(24,27)(25,28)(26,29)(30,31);(5,7)(16,18,19,24,20,25,26,30)(17,21,23,27,22,28,29,31);(8,11,12,10)(9,14,15,13)(17,22)(21,28)(23,29)(27,31)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18808);