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×#34372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1910 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)(4,5,6,7)(8,9,13,11)(10,14,18,16)(12,19,20,15)(17,22,23,21)(25,29)(26,32)(27,35)(37,39);(9,11)(14,16)(15,19)(21,22)(24,25,28,29)(26,33,32,30)(27,31,35,36)(34,39,38,37);(8,10,13,18)(9,14,11,16)(12,17,20,23)(15,21,19,22)(24,25)(26,30)(27,31)(28,29)(32,33)(34,37)(35,36)(38,39);(8,11,13,9)(10,16,18,14)(12,19,20,15)(17,22,23,21)(24,26,28,32)(25,30,29,33)(27,34,35,38)(31,37,36,39);(8,12)(9,15)(10,17)(11,19)(13,20)(14,21)(16,22)(18,23)(24,27)(25,31)(26,34)(28,35)(29,36)(30,37)(32,38)(33,39)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27360);