Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C3×#33455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5,7,6)(9,14)(11,21)(12,23)(15,27)(16,31)(17,28)(19,34)(20,33)(22,25)(24,26)(29,38)(30,39)(32,37)(35,36);(8,9,13,17)(10,15,18,29)(11,22,24,31)(12,25,23,16)(14,26,28,21)(19,35,32,39)(20,36,33,30)(27,37,38,34);(4,6,7,5)(8,10)(9,15)(11,19)(12,20)(13,18)(14,27)(16,30)(17,29)(21,34)(22,35)(23,33)(24,32)(25,36)(26,37)(28,38)(31,39);(10,18)(15,29)(19,32)(20,33)(27,38)(30,36)(34,37)(35,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26443);