Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#8986 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #120 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(5,7)(8,9,13,14,12,10,11,15)(16,17)(18,24,26,31)(19,21)(20,23)(22,25,29,28)(27,30);(4,5,6,7)(8,10,12,9)(11,15,13,14)(17,21)(20,27)(22,29)(25,28);(8,11,12,13)(9,14,10,15)(16,18)(17,22)(19,26)(20,28)(21,29)(23,31)(24,30)(25,27)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1974);