Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#30407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G | |
980 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)(10,14)(11,13)(12,15)(16,17,21,25)(18,26,19,28)(20,30,31,24)(22,29,23,27);(4,5)(6,7)(8,10)(9,12)(11,15)(13,14);(5,7)(16,18,21,19)(17,22,25,23)(20,29,31,27)(24,28,30,26);(9,13)(10,15)(16,19,21,18)(17,23,25,22)(20,27,31,29)(24,26,30,28)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23395);