Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#34365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(4,5,6,7)(9,12)(10,14)(17,21)(19,27)(22,24)(25,31);(9,12)(13,15)(16,17,20,21)(18,24,26,22)(19,23,27,28)(25,30,31,29);(8,9,11,12)(10,13,14,15);(8,10)(9,13)(11,14)(12,15)(16,18,20,26)(17,22,21,24)(19,25,27,31)(23,29,28,30);(16,19)(17,23)(18,25)(20,27)(21,28)(22,29)(24,30)(26,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27353);