Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,14,19)(10,20,23,38)(11,15,13,17)(12,25,28,35)(16,31,34,21)(18,24,36,26)(22,32,29,37)(27,33,39,30);(9,15)(10,21)(11,13)(12,26)(16,32)(17,19)(18,36)(20,30)(22,27)(23,31)(24,28)(29,39)(33,38)(34,37);(5,7)(8,10,11,22,14,23,13,29)(9,16,17,33,19,34,15,30)(12,27,24,21,28,39,26,31)(18,32,35,20,36,37,25,38)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6402);