Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5,7,6)(9,13)(10,16)(15,20)(17,22)(24,25)(26,29)(27,32)(28,33)(30,34)(31,36)(35,37)(38,39);(9,13)(14,19)(15,20)(21,23)(25,26)(27,34)(31,33)(35,39);(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23)(24,26,29,25)(27,32,34,30)(28,33,36,31)(35,38,39,37);(4,6,7,5)(8,10)(9,14)(11,17)(12,16)(13,19)(15,21)(18,22)(20,23)(24,27)(25,30)(26,32)(28,35)(29,34)(31,37)(33,38)(36,39);(8,11,12,18)(9,15,13,20)(10,17,16,22)(14,21,19,23)(24,28,29,36)(25,31,26,33)(27,35,34,39)(30,37,32,38)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27489);