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×#34390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2118 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)(10,12)(11,14)(13,15)(17,21)(18,24)(23,28)(25,30);(9,10)(13,14)(17,21)(22,27)(23,28)(29,31);(8,9,12,10)(11,13,15,14)(16,17)(18,22)(19,23)(20,21)(24,27)(25,29)(26,28)(30,31);(8,10,12,9)(11,14,15,13)(16,18)(17,22)(19,25)(20,24)(21,27)(23,29)(26,30)(28,31);(8,11,12,15)(9,13,10,14)(16,19,20,26)(17,23,21,28)(18,25,24,30)(22,29,27,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27378);