Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C3×#60086 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| prime radical | C2×C3×#585 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
4874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5,6,7)(8,9)(10,19)(11,16)(12,17)(13,26)(14,18)(15,21)(20,33)(22,29)(23,31)(24,32)(25,36)(27,30)(28,35)(34,39)(37,38);(8,10)(9,15)(11,20)(12,21)(13,27)(14,22)(16,28)(17,29)(18,30)(19,26)(23,34)(24,35)(25,37)(31,38)(32,39)(33,36);(8,11)(9,16)(10,20)(12,24)(13,25)(14,23)(15,28)(17,32)(18,31)(19,33)(21,35)(22,34)(26,36)(27,37)(29,39)(30,38);(11,23)(16,31)(20,34)(24,25)(28,38)(32,36)(33,39)(35,37);(8,12,14,13)(9,17,18,26)(10,21,22,27)(11,24,23,25)(15,29,30,19)(16,32,31,36)(20,35,34,37)(28,39,38,33)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53074);