Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#33409 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| prime radical | C24×C3 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5,6,7)(8,9)(10,19)(11,16)(12,26)(13,28)(14,18)(15,23)(17,20)(21,32)(22,33)(24,38)(25,31)(27,39)(29,36)(30,37)(34,35);(8,10,20,35,13,24,27,36,14,21,25,37,11,22,12,23)(9,15,26,33,16,30,31,32,18,29,39,38,28,34,17,19);(10,21)(15,29)(19,32)(22,24)(23,36)(30,34)(33,38)(35,37);(8,11,14,13)(9,16,18,28)(10,22,21,24)(12,25,27,20)(15,30,29,34)(17,26,31,39)(19,33,32,38)(23,37,36,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26397);