Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×#362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #102 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C24×C3 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(37,38,39);(3,4);(5,6,9,13,10,15,25,14)(7,16,18,29,19,33,24,32)(8,20,22,12,23,30,36,31)(11,21,26,17,28,34,27,35);(5,7,10,19)(6,11,15,28)(8,21,23,34)(9,24,25,18)(12,29,31,32)(13,27,14,26)(16,20,33,30)(17,22,35,36)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26378);