Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#20736 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C22×D4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G | |
1358 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)(4,5,6,7)(8,9)(10,13)(11,12)(14,15)(16,17)(18,21)(19,22)(20,24)(23,27)(25,28)(26,30)(29,31);(9,12)(13,15)(16,18,20,26)(17,21,24,30)(19,25,27,31)(22,28,23,29);(17,22)(19,27)(21,28)(23,24)(25,31)(29,30);(8,10,11,14)(9,13,12,15)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13724);