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) = C28 |
| direct product | C3×#34371 |
| 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)(8,9,12,10)(11,14,15,13)(17,22)(18,19)(20,28)(30,31);(9,10)(13,14)(16,17,21,22)(18,24,19,23)(20,25,28,29)(26,31,27,30);(16,18,21,19)(17,23,22,24)(20,26,28,27)(25,30,29,31);(8,10,12,9)(11,14,15,13)(16,19,21,18)(17,24,22,23)(20,27,28,26)(25,31,29,30);(8,11)(9,13)(10,14)(12,15)(16,20)(17,25)(18,26)(19,27)(21,28)(22,29)(23,30)(24,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27359);